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 | |
1279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(6,8)(9,10,14,20)(11,22,25,17)(12,27,29,62)(13,19,16,37)(15,34,33,64)(18,43,45,30)(21,32,47,36)(23,51,53,60)(24,42,31,40)(26,58,57,66)(28,61,38,48)(35,44,65,46)(39,54,49,70)(41,56,50,63)(52,71,55,68)(59,67,72,69);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,23)(13,31)(14,25)(15,26)(16,24)(18,39)(19,42)(20,22)(21,41)(27,60)(28,55)(29,53)(30,54)(32,63)(33,57)(34,66)(35,59)(36,56)(37,40)(38,52)(43,70)(44,69)(45,49)(46,67)(47,50)(48,68)(51,62)(58,64)(61,71)(65,72);(9,12)(10,18)(11,23)(13,28)(14,29)(15,35)(16,38)(17,39)(19,44)(20,45)(21,48)(22,49)(24,52)(25,53)(26,59)(27,34)(30,36)(31,55)(32,43)(33,65)(37,46)(40,67)(41,68)(42,69)(47,61)(50,71)(51,58)(54,56)(57,72)(60,66)(62,64)(63,70);(2,5)(6,8)(9,13,14,16)(10,19,20,37)(11,24,25,31)(12,28,29,38)(15,32,33,36)(17,40,22,42)(18,44,45,46)(21,34,47,64)(23,52,53,55)(26,56,57,63)(27,61,62,48)(30,35,43,65)(39,67,49,69)(41,58,50,66)(51,71,60,68)(54,72,70,59);(1,3,4,7)(2,6,5,8)(11,25)(17,22)(23,53)(24,31)(26,57)(39,49)(40,42)(41,50)(51,60)(52,55)(54,70)(56,63)(58,66)(59,72)(67,69)(68,71)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,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, 7531230);