Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 | |
2020 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)(11,21)(12,25)(13,18)(14,17)(15,20)(16,23)(19,26)(22,36)(24,35)(27,40)(28,31)(29,34)(30,38)(32,39)(33,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,15)(13,27)(14,23)(17,21)(18,32)(19,25)(20,35)(22,37)(24,26)(28,39)(29,38)(30,33)(31,40)(34,36);(3,8)(5,7)(9,12,26,23,14,24,15,11)(10,16,19,35,17,21,20,25)(13,27,37,38,28,39,29,22)(18,30,33,32,31,36,34,40);(2,6)(5,7)(10,17)(13,28)(16,21)(19,20)(22,38)(25,35)(27,39)(29,37);(9,13,26,37,14,28,15,29)(10,18,19,33,17,31,20,34)(11,22,12,27,23,38,24,39)(16,30,35,32,21,36,25,40)Polycyclic presentation:
2^9|c^2=fh,e^2=fh,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=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717243);