Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
876 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)(3,7)(4,5)(6,8)(10,15,18,28)(11,19)(12,24,26,27)(16,20,33,30)(21,37,35,39)(22,23)(32,38)(36,40);(9,10)(11,20)(12,25)(13,28)(14,18)(15,29)(16,23)(17,24)(19,30)(21,38)(22,33)(26,31)(27,34)(32,35)(36,39)(37,40);(2,5)(6,7)(9,11,13,22,14,19,29,23)(10,16,28,20,18,33,15,30)(12,21,27,37,26,35,24,39)(17,32,25,36,34,38,31,40);(1,3,4,8)(2,6,5,7)(9,12,14,26)(10,17,18,34)(11,21,19,35)(13,27,29,24)(15,31,28,25)(16,32,33,38)(20,36,30,40)(22,37,23,39)Polycyclic presentation:
2^9|a^2=h,c^2=eh,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296878);