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 | #42674 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1717 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)(23,27,32,40)(24,29,35,44)(25,28,37,31)(33,39,45,51)(34,41,46,38)(36,42,48,43)(47,50,52,49);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,32)(22,27,30,40)(24,33,35,45)(25,34,37,46)(28,41,31,38)(29,39,44,51)(36,47,48,52)(42,50,43,49);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(22,28)(23,32)(24,35)(25,37)(27,38)(29,43)(30,31)(39,50)(40,41)(42,44)(47,52)(49,51);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24)(22,29)(23,33)(25,36)(26,35)(27,39)(28,42)(30,44)(31,43)(32,45)(34,47)(37,48)(38,49)(40,51)(41,50)(46,52);(6,12)(7,14)(11,20)(13,15)(22,30)(23,32)(24,35)(28,31)(34,46)(36,48)(39,51)(49,50)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=hi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7011691);