Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,15,21,33,19)(11,22,25,16,27,35,26,36)(12,17,24,37,29,38,28,34)(13,30,31,18,32,39,23,40);(1,3,4,7)(2,5,6,8)(9,11,14,25,15,27,33,26)(10,16,19,22,21,36,20,35)(12,13,24,31,29,32,28,23)(17,18,34,30,38,40,37,39);(2,6)(5,8)(9,12,14,24,15,29,33,28)(10,17,20,37,21,38,19,34)(11,23,25,13,27,31,26,32)(16,30,35,18,36,39,22,40);(9,13,14,31,15,32,33,23)(10,18,19,30,21,40,20,39)(11,24,25,29,27,28,26,12)(16,34,22,38,36,37,35,17)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281189);