Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10,12,14)(11,13,15,16)(17,18,20,24,21,25,30,23)(19,26,27,31,29,32,28,22);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,19)(18,22)(20,27)(21,29)(23,32)(24,26)(25,31)(28,30);(2,7)(6,8)(9,11,12,15)(10,13,14,16);(1,4,5,3)(2,8,7,6)(18,23)(19,27)(20,30)(22,31)(24,25)(28,29)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=g,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 360830);