Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,30)(23,27,34,40)(24,29,35,44)(25,28,37,31)(32,39,45,51)(33,38,47,41)(36,42,48,43)(46,49,52,50);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,34)(22,27,30,40)(24,32,35,45)(25,33,37,47)(28,38,31,41)(29,39,44,51)(36,46,48,52)(42,49,43,50);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(22,28)(24,35)(25,37)(27,38)(29,43)(30,31)(32,45)(33,47)(39,50)(40,41)(42,44)(49,51);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24)(22,29)(23,32)(25,36)(26,35)(27,39)(28,42)(30,44)(31,43)(33,46)(34,45)(37,48)(38,49)(40,51)(41,50)(47,52);(6,12)(7,14)(11,20)(13,15)(22,30)(24,35)(27,40)(28,31)(32,45)(36,48)(38,41)(46,52)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=h,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011767);