Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 |
| prime radical | #61075 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2032 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)(3,5)(4,6)(7,8)(10,16)(11,20)(13,14)(18,30)(19,26)(21,35)(22,28)(23,34)(24,27)(29,32)(36,39)(37,38);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,23)(13,26)(14,16)(15,19)(18,24)(20,32)(21,36)(22,33)(25,30)(27,34)(28,29)(31,37)(35,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11,15,22)(10,17,19,33)(12,21,25,38)(13,20,14,28)(16,29,26,32)(18,31,34,40)(23,36,30,39)(24,37,27,35);(3,8)(5,7)(9,12)(10,18)(11,21)(13,27)(14,24)(15,25)(16,30)(17,31)(19,34)(20,35)(22,38)(23,26)(28,37)(29,39)(32,36)(33,40);(2,6)(5,7)Polycyclic presentation:
2^9|c^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962107);