Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2150 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)(11,16)(12,24)(13,18)(14,30)(15,19)(17,27)(20,34)(21,32)(22,29)(23,33)(25,39)(26,36)(28,35)(31,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,23)(10,16,19,33)(12,25,28,37)(13,29,14,21)(17,34,36,40)(18,22,30,32)(20,26,38,27)(24,39,35,31);(9,12,13,26,15,28,14,27)(10,17,30,35,19,36,18,24)(11,20,29,25,23,38,21,37)(16,31,32,40,33,39,22,34);(2,6)(5,7)(9,13,15,14)(10,18,19,30)(11,21,23,29)(12,26,28,27)(16,32,33,22)(17,24,36,35)(20,37,38,25)(31,40,39,34);(3,8)(5,7)(9,10)(11,22)(12,17)(13,30)(14,18)(15,19)(16,29)(20,34)(21,33)(23,32)(24,27)(25,31)(26,35)(28,36)(37,39)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589862);