Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,19)(20,21,24,23);(9,11,12,16)(10,13,14,15)(17,19,21,24,22,18,23,20);(2,4)(11,16)(13,15);(10,14)(13,15)(17,20,21,19,22,24,23,18)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264289, #72264306, #72264332, … |
| maximal normal subgroup of | #72264289, #72264306, #72264332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270698);