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 | |
1960 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,25)(13,18)(14,29)(15,19)(16,22)(17,28)(21,38)(23,30)(24,33)(26,32)(27,36)(31,35)(34,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11,13,22)(10,16,18,20)(12,26,27,40)(14,24,15,30)(17,34,35,38)(19,33,29,23)(21,31,39,28)(25,37,36,32);(9,12,15,31,13,27,14,28)(10,17,29,36,18,35,19,25)(11,21,30,26,22,39,24,40)(16,32,23,38,20,37,33,34);(2,6)(5,7)(10,18)(11,22)(17,35)(19,29)(21,39)(24,30)(25,36)(26,40);(3,8)(5,7)(9,10)(11,23)(12,17)(13,18)(14,19)(15,29)(16,24)(20,30)(21,38)(22,33)(25,28)(26,37)(27,35)(31,36)(32,40)(34,39)Polycyclic presentation:
2^9|b^2=f,c^2=gi,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, 6590472);