Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,21,35,22,24,23,27)(19,28,37,25,31,40,30,38)(20,32,36,42,33,26,34,41)(29,44,46,48,45,43,47,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(18,24)(20,33)(25,38)(27,35)(28,40)(29,45)(34,36)(46,47);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,25)(20,29)(21,30)(22,31)(23,37)(24,38)(26,39)(27,40)(28,35)(32,48)(33,45)(34,46)(36,47)(41,43)(42,44);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20,22,33)(18,26,24,32)(19,29,31,45)(21,36,23,34)(25,39,38,48)(27,42,35,41)(28,43,40,44)(30,47,37,46);(2,6)(5,8)(17,21,22,23)(18,27,24,35)(19,30,31,37)(20,34,33,36)(25,40,38,28)(26,41,32,42)(29,46,45,47)(39,43,48,44)Polycyclic presentation:
2^9|a^2=fg,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6542742);