Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574086 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1796 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)(4,8)(5,6)(9,10,15,14)(11,22,26,17)(12,29,13,19)(16,21,34,20)(18,32,30,31)(23,37,39,40)(24,38,33,35)(25,27,36,28);(2,6)(7,8)(9,11,15,26)(10,17,14,22)(12,24,13,33)(16,27,34,28)(18,37,30,40)(19,38,29,35)(20,36,21,25)(23,31,39,32);(9,12,16,31,15,13,34,32)(10,18,20,29,14,30,21,19)(11,23,27,24,26,39,28,33)(17,35,36,40,22,38,25,37);(1,3,5,4)(2,7,6,8)(9,13,15,12)(10,19,14,29)(11,24,26,33)(16,32,34,31)(17,35,22,38)(18,21,30,20)(23,27,39,28)(25,37,36,40);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,25,26,36)(12,30,13,18)(16,21,34,20)(17,28,22,27)(19,32,29,31)(23,37,39,40)(24,35,33,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492261);