Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,15,14)(11,20,24,16)(12,26,31,36)(13,19,28,18)(17,29,30,32)(21,35,39,40)(22,34,25,23)(27,37,38,33);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,27)(13,22)(14,20)(15,24)(17,35)(18,34)(19,23)(21,29)(25,28)(26,33)(30,40)(31,38)(32,39)(36,37);(2,5)(6,7)(9,12,28,29,15,31,13,32)(10,17,19,36,14,30,18,26)(11,21,25,38,24,39,22,27)(16,33,23,35,20,37,34,40);(1,3,4,8)(2,6,5,7)(9,13,15,28)(10,18,14,19)(11,22,24,25)(12,29,31,32)(16,34,20,23)(17,36,30,26)(21,38,39,27)(33,35,37,40);(9,14,15,10)(11,23,24,34)(12,30,31,17)(13,19,28,18)(16,25,20,22)(21,35,39,40)(26,32,36,29)(27,33,38,37)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,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,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592179);