Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,6,4,7)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,28,30,32)(20,25,22,31)(21,24,29,26);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,23,30)(18,24,27,26)(20,21,22,29)(25,32,31,28);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,23,22)(18,25,27,31)(19,29,30,21)(24,28,26,32);(1,4,5,3)(2,7,8,6)(17,21,22,30,23,29,20,19)(18,26,25,32,27,24,31,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 350574);