Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
1745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,15,21)(11,22,24,17)(12,25,28,36)(13,30,16,19)(14,20,32,40)(18,27,38,29)(23,35,33,37)(26,39,31,34);(1,2)(3,6)(4,8)(5,7)(9,11,15,24)(10,17,21,22)(12,26,28,31)(13,23,16,33)(14,29,32,27)(18,20,38,40)(19,35,30,37)(25,34,36,39);(9,12,15,28)(10,18,21,38)(11,14,24,32)(13,27,16,29)(17,34,22,39)(19,36,30,25)(20,37,40,35)(23,31,33,26);(1,3,5,4)(2,6,7,8)(9,13,15,16)(10,19,21,30)(11,23,24,33)(12,27,28,29)(14,31,32,26)(17,35,22,37)(18,36,38,25)(20,39,40,34);(1,4,5,3)(2,6,7,8)(9,14,15,32)(10,20,21,40)(11,12,24,28)(13,26,16,31)(17,36,22,25)(18,37,38,35)(19,34,30,39)(23,29,33,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=f,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,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583724);