Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#550 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,19,21,20,22,23,24);(1,2,5,6)(3,7,4,8)(9,11,13,12)(10,14,16,15);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(14,16);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=e,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,f^c=fh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 131740);