Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,33,41)(19,25,30,40,29,23,34,36)(20,24,32,39,31,38,44,48)(28,37,43,47,42,35,45,46);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,26)(27,36)(31,42)(32,43)(33,34)(37,38)(39,46)(40,41)(44,45)(47,48);(10,14)(11,15)(17,20,22,32,21,31,33,44)(18,24,27,39,26,38,41,48)(19,28,30,43,29,42,34,45)(23,35,36,46,25,37,40,47);(3,8)(5,7)(10,14)(13,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,37)(27,40)(31,42)(32,43)(33,34)(35,38)(36,41)(39,47)(44,45)(46,48);(2,6)(5,7)(19,29)(23,25)(28,42)(30,34)(35,37)(36,40)(43,45)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dfi,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319641);