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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1725 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)(21,22,26,30)(23,27,34,43)(24,28,36,45)(25,29,38,47)(31,40,33,42)(32,41,49,51)(35,44,50,52)(37,46,39,48);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,34)(22,27,30,43)(24,35,36,50)(25,31,38,33)(28,44,45,52)(29,40,47,42)(32,39,49,37)(41,48,51,46);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(23,31)(25,38)(27,40)(29,47)(32,35)(33,34)(37,39)(41,44)(42,43)(46,48)(49,50)(51,52);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(24,36)(28,45)(32,49)(35,50)(37,39)(41,51)(44,52)(46,48);(7,17)(11,14)(15,16)(19,20)(21,24,26,36)(22,28,30,45)(23,32,34,49)(25,39,38,37)(27,41,43,51)(29,48,47,46)(31,35,33,50)(40,44,42,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=f,e^2=i,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, 6591795);