Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,31)(23,32,27,28)(24,35,38,43)(25,39,41,30)(29,33,46,36)(34,44,42,48)(37,52,40,51)(45,50,47,49);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,26,27)(22,28,31,32)(24,36,38,33)(25,34,41,42)(29,35,46,43)(30,44,39,48)(37,50,40,49)(45,51,47,52);(2,4)(21,24,27,36,26,38,23,33)(22,29,32,35,31,46,28,43)(25,40,42,49,41,37,34,50)(30,47,48,52,39,45,44,51);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,41)(22,30,31,39)(23,34,27,42)(24,37,38,40)(28,44,32,48)(29,45,46,47)(33,49,36,50)(35,51,43,52);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(25,41)(30,39)(34,42)(37,40)(44,48)(45,47)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593695);