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 | |
1965 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)(11,23)(12,28)(13,22)(14,20)(15,21)(16,19)(17,26)(18,30)(24,38)(25,35)(27,37)(29,40)(31,36)(32,39)(33,34);(2,5)(3,7)(9,11)(10,17)(12,29)(13,25)(14,30)(15,26)(16,27)(18,38)(19,35)(20,39)(21,23)(22,37)(24,32)(28,34)(31,33)(36,40);(9,12,15,31)(10,18,21,39)(11,24,26,14)(13,30,16,32)(17,34,23,40)(19,28,22,36)(20,37,38,35)(25,29,27,33);(2,5)(6,8)(9,13,15,16)(10,19,21,22)(11,25,26,27)(12,30,31,32)(14,33,24,29)(17,35,23,37)(18,28,39,36)(20,40,38,34);(1,3,4,7)(2,6,5,8)(9,14)(10,20)(11,12)(13,29)(15,24)(16,33)(17,36)(18,37)(19,34)(21,38)(22,40)(23,28)(25,32)(26,31)(27,30)(35,39)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,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, 6580351);