Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| 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 | |
1979 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)(3,8)(4,6)(5,7)(9,10,14,11)(12,18,26,22)(13,19,30,23)(15,24,16,20)(17,32,21,27)(25,35,39,38)(28,31,37,33)(29,36,40,34);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,28)(15,24)(16,20)(17,32)(19,33)(21,27)(22,26)(23,31)(25,36)(29,35)(30,37)(34,39)(38,40);(9,12,16,32,14,26,15,27)(10,17,24,22,11,21,20,18)(13,25,33,29,30,39,31,40)(19,34,37,38,23,36,28,35);(1,4,5,3)(2,6,7,8)(9,12,14,26)(10,18,11,22)(13,29,30,40)(15,27,16,32)(17,20,21,24)(19,36,23,34)(25,33,39,31)(28,38,37,35);(9,13)(10,19)(11,23)(12,25)(14,30)(15,31)(16,33)(17,34)(18,35)(20,28)(21,36)(22,38)(24,37)(26,39)(27,40)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^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=eh,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, 6491533);