Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1814 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)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,2)(3,4)(5,7,8,6)(9,11,12,16)(10,13,14,15);(2,4)(6,7)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(11,16)(13,15)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328900);