Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #164127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10,11,14,12,13,15,16)(18,21)(19,23)(22,24);(1,2)(3,6)(4,5)(7,8);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,18,20,22)(19,24,23,21);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,c^2=ei,f^2=i,g^2=hi,b^a=be,c^a=cf,f^a=fh,d^b=de,f^c=fi,g^c=ghi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 130365);