Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60762 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,26)(13,19)(14,31)(15,20)(16,21)(18,27)(22,38)(23,34)(24,36)(25,37)(28,39)(29,40)(30,33)(32,35);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,13)(14,24)(15,32)(16,25)(18,38)(19,26)(20,35)(21,37)(22,27)(23,29)(28,30)(31,36)(33,39)(34,40);(9,12,15,27,16,28,14,23)(10,18,20,39,21,34,31,26)(11,22,24,13,25,29,32,30)(17,33,36,38,37,19,35,40);(2,6)(5,7);(3,8)(5,7)(9,13,15,29,16,30,14,22)(10,19,20,40,21,33,31,38)(11,23,24,12,25,27,32,28)(17,34,36,26,37,18,35,39)Polycyclic presentation:
2^9|c^2=h,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714011);