Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33716 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,15,11,17,25,16)(7,18,23,33,20,29,36,34)(8,13,9,24,19,14,26,32)(12,27,30,35,28,21,31,22);(5,7,19,27,10,23,26,35,11,20,8,21,25,36,9,22)(6,12,24,34,16,31,13,29,17,28,32,33,15,30,14,18);(1,2,3,4)(7,20)(12,28)(18,29)(21,27)(22,35)(23,36)(30,31)(33,34);(5,8,10,9,11,19,25,26)(6,13,15,24,17,14,16,32)(7,21,23,22,20,27,36,35)(12,29,30,34,28,18,31,33)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=f,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403309);