Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,27,32)(23,28,36,44)(24,29,39,47)(25,30,26,31)(33,41,50,52)(34,42,35,43)(37,45,49,51)(38,46,40,48);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,27,36)(22,28,32,44)(24,37,39,49)(25,34,26,35)(29,45,47,51)(30,42,31,43)(33,40,50,38)(41,48,52,46);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,27,39)(22,29,32,47)(23,33,36,50)(25,38,26,40)(28,41,44,52)(30,46,31,48)(34,49,35,37)(42,51,43,45);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25,27,26)(22,30,32,31)(23,34,36,35)(24,38,39,40)(28,42,44,43)(29,46,47,48)(33,49,50,37)(41,51,52,45);(7,17)(11,14)(15,16)(19,20)(23,35)(25,26)(28,43)(30,31)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fi,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, 6591800);