Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2206 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,7,8);(1,2,4,3)(10,14)(11,17)(12,19)(15,22)(16,23)(18,21)(20,24);(6,8)(9,10)(11,18)(12,21)(13,15)(14,17)(16,19)(20,23)(22,24);(1,3,4,2)(10,15)(14,22)(16,21)(18,23)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319120, #72319493, #72319542 |
| maximal normal subgroup of | #72319120, #72319493, #72319542 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408803);