Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,23)(21,31)(22,26)(24,29)(25,28)(30,32);(9,11,12,16)(10,13,14,15)(17,19,28,30,22,29,21,20)(18,23,31,24,26,32,25,27);(1,2,3,4)(11,16)(13,15);(10,14)(13,15)(17,20,21,29,22,30,28,19)(18,24,25,23,26,27,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=f,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271001);