Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| socle | C23 | C22×C22⋊C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(1,2,4,3)(5,6)(7,8);(1,3,4,2)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72313971, #72314026 |
| maximal normal subgroup of | #72313971, #72314026 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265177);