Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| 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 | |
1366 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)(3,7)(9,10,13,18)(11,21,23,16)(12,26,28,55)(14,33,31,61)(15,34,32,20)(17,38,39,29)(19,42,41,27)(22,46,48,54)(24,53,51,62)(25,37,52,45)(30,60,59,57)(35,49,43,64)(36,47,44,66)(40,58,56,65)(50,71,70,68)(63,72,67,69);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,35)(18,21)(19,36)(20,37)(26,54)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,62)(34,45)(38,64)(39,43)(40,63)(41,44)(42,66)(46,55)(53,61)(56,67)(57,71)(58,69)(59,70)(60,68)(65,72);(9,12,27,58,15,30,31,38,13,28,42,65,32,59,14,29)(10,17,33,57,20,40,41,55,18,39,61,60,34,56,19,26)(11,22,47,69,25,50,51,64,23,48,66,72,52,70,24,49)(16,35,62,71,37,63,44,46,21,43,53,68,45,67,36,54);(2,5)(6,8)(11,23)(16,21)(22,48)(24,51)(25,52)(35,43)(36,44)(37,45)(46,54)(47,66)(49,64)(50,70)(53,62)(63,67)(68,71)(69,72);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(17,39)(19,41)(20,34)(22,48)(24,51)(25,52)(26,55)(33,61)(40,56)(47,66)(49,64)(50,70)(57,60)(69,72)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532191);