Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,21,18,16,25,20)(15,19,23,27,24,22,28,26);(6,9)(10,12)(13,15,17,23,18,24,25,28)(14,19,21,27,16,22,20,26);(1,2,3,4);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,22)(17,20)(19,24)(21,25)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,g^d=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300316);