Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | #2358 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1095 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)(4,6)(5,8)(9,10)(11,23,26,36)(12,18)(13,30,31,40)(14,20)(15,21)(16,22)(17,24,37,25)(19,32,34,27)(28,38)(29,35)(33,39);(9,11)(10,17)(12,27)(13,28)(14,25)(15,24)(16,26)(18,30)(19,39)(20,36)(21,23)(22,37)(29,32)(31,33)(34,38)(35,40);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,13)(14,28)(15,33)(16,29)(17,34)(19,37)(20,38)(21,39)(22,35)(23,30)(24,27)(25,32)(26,31)(36,40);(2,6)(5,7)(9,13)(10,19)(11,12)(14,32)(15,27)(16,31)(17,35)(18,37)(20,30)(21,40)(22,34)(23,38)(24,33)(25,28)(26,29)(36,39)Polycyclic presentation:
2^9|a^2=h,c^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, 291354);