Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,26,29)(21,24,28,25)(27,31,32,30);(1,2)(3,4)(6,7)(9,11,12,15)(10,13,14,16)(18,22)(23,29)(24,25)(30,31);(2,4)(5,7,8,6)(10,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,29)(24,31)(25,30)(28,32);(6,7)(18,24)(19,26)(21,28)(22,25)(23,30)(29,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363038);