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 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,6)(7,8)(9,10)(11,21)(12,26)(13,18)(14,32)(15,35)(16,23)(17,28)(19,34)(20,31)(22,48)(24,54)(25,39)(27,60)(29,63)(30,59)(33,44)(36,43)(37,66)(38,68)(40,61)(41,64)(42,62)(45,50)(46,56)(47,53)(49,58)(51,69)(52,65)(55,67)(57,71)(70,72);(1,2,5,6)(3,7,4,8)(9,11,13,23)(10,16,18,21)(12,22,27,49)(14,24,30,52)(15,25,31,53)(17,37,40,45)(19,38,42,46)(20,39,35,47)(26,58,60,48)(28,50,61,66)(29,51,36,57)(32,65,59,54)(33,55,64,72)(34,56,62,68)(41,67,44,70)(43,69,63,71);(1,3,5,4)(2,7,6,8)(9,12,13,27)(10,17,18,40)(11,22,23,49)(14,33,30,64)(15,36,31,29)(16,37,21,45)(19,43,42,63)(20,44,35,41)(24,55,52,72)(25,57,53,51)(26,59,60,32)(28,62,61,34)(38,69,46,71)(39,70,47,67)(48,65,58,54)(50,68,66,56);(1,4,5,3)(2,7,6,8)(11,23)(16,21)(22,49)(24,52)(25,53)(37,45)(38,46)(39,47)(48,58)(50,66)(51,57)(54,65)(55,72)(56,68)(67,70)(69,71);(10,18)(11,23)(17,40)(19,42)(20,35)(22,49)(24,52)(25,53)(26,60)(32,59)(41,44)(43,63)(50,66)(51,57)(55,72)(56,68)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=fi,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, 7531178);