Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| 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 | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1756 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)(3,4)(5,7)(6,8)(9,10,14,19,15,12,25,18)(11,16,22,33,24,26,23,20)(13,28,30,27,31,34,29,17)(21,35,37,39,38,40,36,32);(1,3)(2,6)(4,7)(5,8)(9,11,15,24)(10,16,12,26)(13,21,31,38)(14,22,25,23)(17,32,27,39)(18,20,19,33)(28,35,34,40)(29,36,30,37);(1,4,5,6)(2,3,7,8)(9,12)(10,15)(11,20)(13,27)(14,19)(16,23)(17,31)(18,25)(21,35)(22,26)(24,33)(28,30)(29,34)(32,37)(36,39)(38,40);(2,7)(3,8)(9,13)(10,17)(11,21)(12,27)(14,29)(15,31)(16,32)(18,28)(19,34)(20,35)(22,36)(23,37)(24,38)(25,30)(26,39)(33,40);(9,14,15,25)(10,18,12,19)(11,22,24,23)(13,29,31,30)(16,20,26,33)(17,28,27,34)(21,36,38,37)(32,35,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410662);