Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(17,18,21,27,22,26,31,25)(19,28,30,23,29,32,20,24);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(19,29)(20,30)(23,24)(28,32);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,19,21,30,22,29,31,20)(18,23,25,28,26,24,27,32);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20,31,29,22,30,21,19)(18,24,25,32,26,23,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,d^2=gh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286861);