Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#466 |
| 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 | |
1872 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)(3,17,24,58)(4,25,29,53)(5,12,30,47)(6,34,36,51)(7,15,8,14)(10,38,43,21)(11,37,46,28)(13,35,52,18)(19,62,66,106)(20,55,67,103)(22,57,23,56)(26,74,76,99)(27,79,80,45)(31,82,81,98)(32,50,33,49)(39,88,90,107)(40,87,91,68)(41,71,42,59)(44,84,95,78)(48,83,97,61)(54,72,102,65)(60,75,85,100)(63,110,111,127)(64,73,86,94)(69,105,70,104)(77,119,118,96)(89,122,123,128)(92,114,93,108)(101,115,126,113)(109,117,120,125)(112,116,121,124);(1,3,18,59,8,23,36,38,9,24,35,71,7,22,6,21)(2,10,34,57,15,42,52,58,16,43,51,56,14,41,13,17)(4,19,60,107,28,65,80,86,29,66,85,88,37,72,27,64)(5,20,61,108,33,70,81,87,30,67,83,114,32,69,31,68)(11,39,75,62,25,73,79,102,46,90,100,106,53,94,45,54)(12,40,82,105,50,93,97,103,47,91,98,104,49,92,48,55)(26,63,109,128,78,113,118,121,76,111,120,122,84,115,77,112)(44,89,117,110,74,116,119,126,95,123,125,127,99,124,96,101);(1,4)(2,11)(3,19)(5,26)(6,27)(7,37)(8,28)(9,29)(10,39)(12,44)(13,45)(14,53)(15,25)(16,46)(17,54)(18,60)(20,63)(21,64)(22,72)(23,65)(24,66)(30,76)(31,77)(32,84)(33,78)(34,75)(35,85)(36,80)(38,86)(40,89)(41,94)(42,73)(43,90)(47,95)(48,96)(49,99)(50,74)(51,100)(52,79)(55,101)(56,106)(57,62)(58,102)(59,107)(61,109)(67,111)(68,112)(69,115)(70,113)(71,88)(81,118)(82,117)(83,120)(87,121)(91,123)(92,124)(93,116)(97,119)(98,125)(103,126)(104,127)(105,110)(108,128)(114,122);(1,5)(2,12)(3,20)(4,26)(6,31)(7,32)(8,33)(9,30)(10,40)(11,44)(13,48)(14,49)(15,50)(16,47)(17,55)(18,61)(19,63)(21,68)(22,69)(23,70)(24,67)(25,74)(27,77)(28,78)(29,76)(34,82)(35,83)(36,81)(37,84)(38,87)(39,89)(41,92)(42,93)(43,91)(45,96)(46,95)(51,98)(52,97)(53,99)(54,101)(56,104)(57,105)(58,103)(59,108)(60,109)(62,110)(64,112)(65,113)(66,111)(71,114)(72,115)(73,116)(75,117)(79,119)(80,118)(85,120)(86,121)(88,122)(90,123)(94,124)(100,125)(102,126)(106,127)(107,128);(5,30)(12,47)(20,67)(26,76)(31,81)(32,33)(40,91)(44,95)(48,97)(49,50)(55,103)(61,83)(63,111)(68,87)(69,70)(74,99)(77,118)(78,84)(82,98)(89,123)(92,93)(96,119)(101,126)(104,105)(108,114)(109,120)(110,127)(112,121)(113,115)(116,124)(117,125)(122,128)Polycyclic presentation:
2^9|a^2=i,b^2=fh,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,h^a=hi,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530189);