Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | #61196 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,19)(12,26)(13,31)(14,22)(15,21)(16,20)(17,25)(18,30)(23,38)(24,33)(27,35)(28,37)(29,36)(32,40)(34,39);(1,3,5,4)(2,6,7,8)(9,11,15,25)(10,17,21,19)(12,27,29,40)(13,14,24,16)(18,34,37,38)(20,33,22,31)(23,28,39,30)(26,32,36,35);(9,12)(10,18)(11,23)(13,27)(14,30)(15,29)(16,28)(17,32)(19,35)(20,26)(21,37)(22,36)(24,40)(25,39)(31,34)(33,38);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,14)(12,27)(15,24)(16,25)(17,21)(18,35)(20,31)(22,33)(23,30)(26,34)(28,39)(29,40)(32,37)(36,38);(2,7)(6,8)(9,14,15,16)(10,20,21,22)(11,24,25,13)(12,28,29,30)(17,33,19,31)(18,36,37,26)(23,27,39,40)(32,34,35,38)Polycyclic presentation:
2^9|b^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,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, 7424819);