Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
1217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,31,36)(27,32,40,48)(28,33,43,51)(29,34,30,35)(37,45,54,56)(38,46,39,47)(41,49,53,55)(42,50,44,52);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,31,40)(26,32,36,48)(28,41,43,53)(29,38,30,39)(33,49,51,55)(34,46,35,47)(37,44,54,42)(45,52,56,50);(2,7)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,33)(27,37)(29,42)(30,44)(31,43)(32,45)(34,50)(35,52)(36,51)(38,53)(39,41)(40,54)(46,55)(47,49)(48,56);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,31,30)(26,34,36,35)(27,38,40,39)(28,42,43,44)(32,46,48,47)(33,50,51,52)(37,53,54,41)(45,55,56,49);(1,4,5,3)(2,8,7,6)(11,21)(15,18)(19,20)(23,24)(27,39)(29,30)(32,47)(34,35)(37,41)(38,40)(42,44)(45,49)(46,48)(50,52)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=f,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6627034);