Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,15,19,16,21,14,20)(11,22,17,12,25,27,24,28)(13,18,31,35,32,36,30,37)(23,38,33,26,39,34,29,40);(2,5)(3,7)(9,11,15,17,16,25,14,24)(10,12,20,22,21,28,19,27)(13,29,30,39,32,33,31,23)(18,34,35,40,36,38,37,26);(1,2)(3,6)(4,5)(7,8)(9,12,16,28)(10,17,21,24)(11,20,25,19)(13,26,32,40)(14,27,15,22)(18,33,36,29)(23,37,39,35)(30,34,31,38);(2,5)(6,8)(9,13,16,32)(10,18,21,36)(11,23,25,39)(12,26,28,40)(14,30,15,31)(17,33,24,29)(19,35,20,37)(22,38,27,34);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,21,20)(11,24,25,17)(12,27,28,22)(13,30,32,31)(18,35,36,37)(23,29,39,33)(26,34,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7056610);