Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| 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 | |
1264 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)(14,33)(15,37)(16,21)(18,30)(20,35)(22,49)(23,40)(24,41)(25,54)(26,57)(28,59)(29,60)(31,61)(32,48)(34,44)(36,64)(38,47)(39,51)(42,56)(43,62)(45,66)(46,63)(50,71)(52,69)(53,72)(55,67)(58,70)(65,68);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,28)(13,23)(14,25)(15,26)(16,24)(18,43)(19,40)(20,42)(21,41)(22,31)(27,59)(29,58)(30,62)(32,52)(33,54)(34,65)(35,56)(36,53)(37,57)(38,50)(39,46)(44,68)(45,67)(47,71)(48,69)(49,61)(51,63)(55,66)(60,70)(64,72);(9,12)(10,18)(11,22)(13,29)(14,34)(15,38)(16,31)(17,39)(19,44)(20,47)(21,46)(23,50)(24,28)(25,55)(26,58)(27,33)(30,35)(32,63)(36,66)(37,45)(40,67)(41,43)(42,70)(48,60)(49,54)(51,56)(52,62)(53,65)(57,68)(59,72)(61,64)(69,71);(2,5)(6,8)(9,13,16,15)(10,19,21,37)(11,23,24,26)(12,29,31,38)(14,32,36,35)(17,40,41,57)(18,44,46,45)(20,33,48,64)(22,50,28,58)(25,52,53,56)(27,60,61,47)(30,34,63,66)(39,67,43,68)(42,54,69,72)(49,71,59,70)(51,55,62,65);(1,3,4,7)(2,6,5,8)(11,24)(17,41)(22,28)(23,26)(25,53)(39,43)(40,57)(42,69)(49,59)(50,58)(51,62)(52,56)(54,72)(55,65)(67,68)(70,71)Polycyclic presentation:
2^9|d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531235);