Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(6,7)(11,15)(13,16)(17,18,22,24,21,26,20,25)(19,23,29,32,27,31,28,30);(5,6,8,7)(9,10,12,14)(11,16,15,13)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,30)(26,31);(2,4)(9,11,12,15)(10,13,14,16)(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(5,7,8,6)(18,25)(20,22)(23,30)(24,26)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fg,b^2=egi,c^2=gi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311043);