Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,17,25,16)(7,12,9,24,19,14,26,28)(8,20,23,33,21,27,36,34)(13,29,31,35,30,18,32,22);(2,4)(5,7,10,9,11,19,25,26)(6,12,15,24,17,14,16,28)(8,18,23,22,21,29,36,35)(13,27,31,34,30,20,32,33);(5,8,19,29,10,23,26,35,11,21,7,18,25,36,9,22)(6,13,24,34,16,32,12,27,17,30,28,33,15,31,14,20);(8,21)(13,30)(18,29)(20,27)(22,35)(23,36)(31,32)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402230);