Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10,12,14)(11,15,16,13)(18,21)(19,22)(23,24);(1,2)(3,4)(17,18,22,24,20,23,19,21);(2,4)(9,11,12,16)(10,13,14,15);(6,8)(18,21)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=ei,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305777, #72305797, #72305807, … |
| maximal normal subgroup of | #72305777, #72305797, #72305807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332037);