Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622850 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#447 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #3573 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 128 points:
(1,2,9,16)(3,17,22,58)(4,23,27,54)(5,12,28,47)(6,32,36,52)(7,15,8,14)(10,39,43,20)(11,37,46,26)(13,34,53,35)(18,59,64,97)(19,57,65,106)(21,68,38,51)(24,70,72,101)(25,75,78,45)(29,80,79,100)(30,50,31,49)(33,55,86,42)(40,76,92,112)(41,89,93,66)(44,84,95,74)(48,82,98,83)(56,63,105,62)(60,107,108,125)(61,69,88,91)(67,113,85,99)(71,87,103,77)(73,117,116,96)(81,102,120,94)(90,118,123,128)(104,111,127,110)(109,114,122,124)(115,121,126,119);(1,3,9,22)(2,10,16,43)(4,18,27,64)(5,19,28,65)(6,33,36,86)(7,21,8,38)(11,40,46,92)(12,41,47,93)(13,51,53,68)(14,42,15,55)(17,52,58,32)(20,34,39,35)(23,69,54,91)(24,60,72,108)(25,76,78,112)(26,62,37,63)(29,81,79,120)(30,67,31,85)(44,90,95,123)(45,97,75,59)(48,99,98,113)(49,94,50,102)(56,103,105,71)(57,100,106,80)(61,87,88,77)(66,82,89,83)(70,114,101,124)(73,118,116,128)(74,110,84,111)(96,125,117,107)(104,126,127,115)(109,121,122,119);(1,4)(2,11)(3,18)(5,24)(6,25)(7,37)(8,26)(9,27)(10,40)(12,44)(13,45)(14,54)(15,23)(16,46)(17,56)(19,60)(20,61)(21,63)(22,64)(28,72)(29,73)(30,84)(31,74)(32,71)(33,76)(34,87)(35,77)(36,78)(38,62)(39,88)(41,90)(42,91)(43,92)(47,95)(48,96)(49,101)(50,70)(51,97)(52,103)(53,75)(55,69)(57,104)(58,105)(59,68)(65,108)(66,109)(67,111)(79,116)(80,115)(81,118)(82,121)(83,119)(85,110)(86,112)(89,122)(93,123)(94,124)(98,117)(99,125)(100,126)(102,114)(106,127)(107,113)(120,128);(1,5)(2,12)(3,19)(4,24)(6,29)(7,30)(8,31)(9,28)(10,41)(11,44)(13,48)(14,49)(15,50)(16,47)(17,57)(18,60)(20,66)(21,67)(22,65)(23,70)(25,73)(26,74)(27,72)(32,80)(33,81)(34,82)(35,83)(36,79)(37,84)(38,85)(39,89)(40,90)(42,94)(43,93)(45,96)(46,95)(51,99)(52,100)(53,98)(54,101)(55,102)(56,104)(58,106)(59,107)(61,109)(62,110)(63,111)(64,108)(68,113)(69,114)(71,115)(75,117)(76,118)(77,119)(78,116)(86,120)(87,121)(88,122)(91,124)(92,123)(97,125)(103,126)(105,127)(112,128);(5,28)(12,47)(19,65)(24,72)(29,79)(30,31)(41,93)(44,95)(48,98)(49,50)(57,106)(60,108)(66,89)(67,85)(70,101)(73,116)(74,84)(80,100)(81,120)(82,83)(90,123)(94,102)(96,117)(99,113)(104,127)(107,125)(109,122)(110,111)(114,124)(115,126)(118,128)(119,121)Polycyclic presentation:
2^9|a^2=i,b^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,h^a=hi,f^b=fh,h^b=hi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530049);