Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(6,8)(9,10)(11,17)(12,27)(13,19)(14,32)(15,36)(16,20)(18,29)(21,34)(22,47)(23,39)(24,41)(25,53)(26,56)(28,58)(30,59)(31,45)(33,42)(35,61)(37,46)(38,50)(40,55)(43,60)(44,62)(48,68)(49,69)(51,67)(52,71)(54,63)(57,66)(64,72)(65,70);(1,2)(3,6)(4,5)(7,8)(9,11,16,24)(10,17,20,41)(12,22,30,49)(13,23,15,26)(14,25,35,52)(18,38,43,65)(19,39,36,56)(21,40,45,67)(27,47,59,69)(28,48,37,57)(29,50,60,70)(31,51,34,55)(32,53,61,71)(33,54,62,72)(42,63,44,64)(46,66,58,68);(9,12)(10,18)(11,22)(13,28)(14,33)(15,37)(16,30)(17,38)(19,42)(20,43)(21,46)(23,48)(24,49)(25,54)(26,57)(27,32)(29,34)(31,60)(35,62)(36,44)(39,63)(40,66)(41,65)(45,58)(47,53)(50,55)(51,70)(52,72)(56,64)(59,61)(67,68)(69,71);(2,5)(6,8)(9,13,16,15)(10,19,20,36)(11,23,24,26)(12,28,30,37)(14,31,35,34)(17,39,41,56)(18,42,43,44)(21,32,45,61)(22,48,49,57)(25,51,52,55)(27,58,59,46)(29,33,60,62)(38,63,65,64)(40,53,67,71)(47,68,69,66)(50,54,70,72);(1,3,4,7)(2,6,5,8)(10,20)(11,24)(18,43)(19,36)(21,45)(22,49)(23,26)(25,52)(27,59)(32,61)(42,44)(46,58)(48,57)(50,70)(51,55)(54,72)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531309);