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)(9,10,15,20)(11,16,22,38)(12,25,29,61)(13,30,34,60)(14,35,32,19)(17,41,44,27)(18,45,47,33)(21,49,26,59)(23,54,52,72)(24,57,53,40)(28,63,36,46)(31,48,65,43)(37,62,42,50)(39,68,67,56)(51,71,58,69)(55,70,64,66);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,22)(17,42)(18,39)(19,40)(20,38)(21,29)(25,59)(27,62)(28,58)(30,54)(31,64)(32,53)(33,56)(34,52)(35,57)(36,51)(37,44)(41,50)(43,70)(45,68)(46,71)(47,67)(48,66)(49,61)(55,65)(60,72)(63,69);(9,12,15,29)(10,17,20,44)(11,21,22,26)(13,31,34,65)(14,36,32,28)(16,37,38,42)(18,46,47,63)(19,48,35,43)(23,55,52,64)(24,58,53,51)(25,60,61,30)(27,45,41,33)(39,69,67,71)(40,70,57,66)(49,72,59,54)(50,68,62,56);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(11,22)(16,38)(21,26)(23,52)(24,53)(37,42)(39,67)(40,57)(49,59)(50,62)(51,58)(54,72)(55,64)(56,68)(66,70)(69,71)Polycyclic presentation:
2^9|a^2=i,c^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, 7531234);