Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| 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 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
752 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)(9,10,14,18)(11,19,17,16)(12,21,15,24)(13,22,23,20);(5,6)(7,8)(9,11)(10,15)(12,18)(13,23)(14,17)(16,20)(19,22);(2,4)(6,8)(9,11,14,17)(10,16,18,19)(12,20,15,22)(13,21,23,24);(2,4)(9,12,21,18,11,20,23,19,14,15,24,10,17,22,13,16)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,h^a=hi,c^b=cg,d^b=de,e^b=eh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72344000 |
| maximal normal subgroup of | #72344000 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416789);