Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,38)(28,33)(29,37)(30,35)(31,36)(32,41)(34,45)(39,53)(40,47)(42,50)(43,52)(44,51)(46,56)(48,55)(49,54);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,31,41)(26,32,36,38)(28,42,44,55)(29,40,45,30)(33,48,51,50)(34,47,37,35)(39,54,56,43)(46,49,53,52);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,29,43,31,44,45,54)(26,33,37,52,36,51,34,49)(27,39,30,42,41,56,40,55)(32,46,47,48,38,53,35,50);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,31,45)(26,34,36,37)(27,40,41,30)(28,43,44,54)(32,47,38,35)(33,49,51,52)(39,55,56,42)(46,48,53,50);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,30,45,27,31,40,29,41)(26,35,34,38,36,47,37,32)(28,42,54,39,44,55,43,56)(33,50,49,53,51,48,52,46)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fhi,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6741262);