Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6,7,8)(17,18)(19,24)(20,29)(21,31)(22,26)(23,27)(25,30)(28,32);(9,10,12,14)(11,15,16,13)(17,19,21,27,22,28,30,20)(18,23,25,24,26,29,31,32);(2,4)(9,11,12,16)(10,13,14,15);(17,20,30,28,22,27,21,19)(18,24,31,23,26,32,25,29)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,d^2=ei,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, 270146);