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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1995 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)(12,24)(13,18)(14,30)(15,19)(16,20)(21,31)(22,32)(23,33)(25,38)(26,34)(27,39)(28,35)(29,36)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11,16,23)(10,17,20,33)(12,14,22,15)(13,26,29,25)(18,34,36,38)(19,24,30,32)(21,27,37,28)(31,39,40,35);(3,8)(5,7)(9,12,15,11,16,22,14,23)(10,17,19,32,20,33,30,24)(13,25,28,21,29,26,27,37)(18,31,35,34,36,40,39,38);(2,6)(5,7);(9,13,14,27,16,29,15,28)(10,18,30,39,20,36,19,35)(11,21,12,25,23,37,22,26)(17,31,24,38,33,40,32,34)Polycyclic presentation:
2^9|b^2=i,c^2=fh,e^2=fhi,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, 6717231);