Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C23 | C22×C42 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
5370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,12,11)(13,14,15,16)(17,18,21,19)(20,22,24,23);(6,8)(13,14,15,16)(18,19)(22,23);(1,2,3,4)(5,6,7,8)(9,11,12,10)(13,14)(15,16)(17,19,21,18)(20,22,24,23);(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de128 conjugacy classes — view the full character table.
| maximal subgroup of | #72330382, #72330447, #72330488, … |
| maximal normal subgroup of | #72330382, #72330447, #72330488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61130);