Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
2019 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,26)(13,31)(14,20)(15,21)(16,19)(17,25)(18,30)(23,38)(24,35)(27,40)(28,39)(29,36)(32,34)(33,37);(2,5)(3,7)(9,11)(10,17)(12,27)(13,24)(14,30)(15,25)(16,33)(18,38)(19,35)(20,39)(21,22)(23,28)(26,34)(29,32)(31,37)(36,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,28)(14,25)(15,29)(16,30)(17,34)(19,36)(20,37)(21,39)(22,40)(24,32)(26,31)(27,33)(35,38);(2,5)(6,8)(9,13,15,16)(10,19,21,31)(11,24,25,33)(12,28,29,30)(14,27,23,32)(17,35,22,37)(18,36,39,26)(20,34,38,40);(9,14,15,23)(10,20,21,38)(11,12,25,29)(13,32,16,27)(17,36,22,26)(18,37,39,35)(19,40,31,34)(24,30,33,28)Polycyclic presentation:
2^9|c^2=fi,d^2=i,e^2=i,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,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, 6576463);