Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10,13,17,14,11,18,16)(12,15,21,23,20,19,24,22);(1,2,3,4)(5,6,8,7)(12,20)(15,19)(21,24)(22,23);(5,7,8,6)(9,11)(10,14)(12,19)(13,16)(15,20)(17,18)(21,22)(23,24);(9,12,13,21,14,20,18,24)(10,15,17,23,11,19,16,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,f^c=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261593);