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 | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16,14,17,12,18)(11,19,20,23,21,15,22,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(10,16)(12,13)(15,23)(17,18)(19,24)(20,22)Polycyclic presentation:
2^9|a^2=h,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^c=hi,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72235097, #72235111, #72251685, … |
| maximal normal subgroup of | #72235097, #72235111, #72251685, … |
| maximal quotient of | #72264234, #72264305 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299455);