Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
1736 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)(3,4)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22)(23,28)(24,29)(25,30)(26,32)(27,31)(33,44)(34,35)(36,46)(37,47)(38,43)(39,41)(40,48)(42,45)(49,51)(50,52);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,27,36)(22,28,31,46)(24,37,40,49)(25,34,32,45)(26,42,30,35)(29,47,48,51)(33,43,50,39)(38,52,41,44);(2,4)(21,24,27,40)(22,29,31,48)(23,33,36,50)(25,41,32,38)(26,43,30,39)(28,44,46,52)(34,51,45,47)(35,49,42,37);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,32)(22,30,31,26)(23,34,36,45)(24,38,40,41)(28,35,46,42)(29,43,48,39)(33,47,50,51)(37,52,49,44);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,31)(23,35)(26,30)(28,45)(29,48)(33,49)(34,46)(36,42)(37,50)(39,43)(44,47)(51,52)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582622);