Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1334 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)(3,5)(4,6)(7,8)(9,10,13,19,14,17,12,18)(11,20,21,24,22,15,23,16);(9,11)(10,15)(12,21)(13,23)(14,22)(16,18)(17,20)(19,24);(2,5)(3,8)(6,7)(9,11)(10,16,17,24)(12,21)(13,23)(14,22)(15,18,20,19);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,c^2=e,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^b=gh,f^c=fi,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232781, #72232818, #72232822, … |
| maximal normal subgroup of | #72232781, #72232818, #72232822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97466);