Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1146 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)(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,23,20,24)(21,25,27,30)(28,31,29,32);(2,4)(5,7,8,6)(10,13)(14,15)(17,19,27,28,22,20,21,29)(18,23,30,31,26,24,25,32);(2,4)(17,20,22,19)(18,24,26,23)(21,29,27,28)(25,32,30,31);(2,4)(9,11,12,16)(10,14,13,15)(17,21,22,27)(18,25,26,30)(19,28,20,29)(23,31,24,32)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275012);