Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499777 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23848 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,17)(4,24)(5,30)(6,34)(7,15)(8,14)(9,16)(10,20)(11,28)(12,31)(13,38)(18,62)(19,43)(21,74)(22,54)(23,61)(25,78)(26,81)(27,86)(29,49)(32,92)(33,93)(35,58)(36,52)(37,57)(39,56)(40,50)(41,55)(42,97)(44,72)(45,73)(46,64)(47,105)(48,90)(51,95)(53,94)(59,65)(60,67)(63,104)(66,110)(68,102)(69,101)(70,103)(71,100)(75,80)(76,82)(77,89)(79,121)(83,91)(84,124)(85,108)(87,99)(88,107)(96,109)(98,116)(106,125)(111,118)(112,119)(113,120)(114,115)(117,128)(122,123)(126,127);(1,3)(2,10)(4,25)(5,19)(6,35)(7,21)(8,22)(9,23)(11,46)(12,43)(13,54)(14,44)(15,58)(16,45)(17,34)(18,28)(20,38)(24,75)(26,79)(27,87)(29,65)(30,60)(31,71)(32,68)(33,70)(36,67)(37,73)(39,72)(40,66)(41,69)(42,49)(47,104)(48,59)(50,99)(51,101)(52,102)(53,103)(55,100)(56,61)(57,74)(62,77)(63,83)(64,90)(76,121)(78,86)(80,89)(81,122)(82,115)(84,117)(85,116)(88,123)(91,114)(92,112)(93,113)(94,119)(95,120)(96,97)(98,106)(105,118)(107,128)(108,111)(109,110)(124,127)(125,126);(1,4)(2,11)(3,18)(5,26)(6,27)(7,40)(8,28)(9,29)(10,42)(12,47)(13,48)(14,49)(15,24)(16,50)(17,59)(19,63)(20,64)(21,25)(22,65)(23,66)(30,91)(31,82)(32,84)(33,85)(34,77)(35,80)(36,88)(37,96)(38,89)(39,90)(41,83)(43,98)(44,99)(45,75)(46,58)(51,106)(52,76)(53,107)(54,110)(55,108)(56,109)(57,86)(60,111)(61,78)(62,74)(67,115)(68,79)(69,116)(70,117)(71,118)(72,97)(73,87)(81,92)(93,124)(94,125)(95,105)(100,127)(101,128)(102,104)(103,121)(112,114)(113,122)(119,123)(120,126);(1,5,9,33)(2,12,16,53)(3,19,23,70)(4,26,29,85)(6,36,39,95)(7,32,8,41)(10,43,45,103)(11,47,50,107)(13,55,57,92)(14,51,15,52)(17,60,61,113)(18,63,66,117)(20,71,73,119)(21,68,22,69)(24,76,49,106)(25,79,65,116)(27,88,90,105)(28,83,40,84)(30,56,93,34)(31,37,94,38)(35,67,72,120)(42,98,75,121)(44,101,58,102)(46,104,99,128)(48,108,86,81)(54,100,74,112)(59,111,78,122)(62,114,110,127)(64,118,87,123)(77,91,109,124)(80,115,97,126)(82,96,125,89)Polycyclic presentation:
2^9|d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406974);