Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454982 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ 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:
(1,2,4,6)(3,5,7,8)(9,10)(11,18,19,14)(12,15)(13,16)(17,21)(20,23,24,22);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,20)(13,19)(15,22)(16,18)(17,23)(21,24);(3,7)(5,8)(12,21)(14,18)(15,17)(20,24);(10,15,16,17)(14,22,18,23)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285055, #72286224, #72286246, … |
| maximal normal subgroup of | #72285055, #72286224, #72286246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374330);