Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184979 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,f^c=fi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257610, #72257614, #72257814, … |
| maximal normal subgroup of | #72257610, #72257614, #72257814, … |
| maximal quotient of | #72264084, #72264409 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299451);