Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 | |
1756 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)(3,4)(5,7)(6,8)(9,10)(11,17)(12,25)(13,27)(14,18)(15,19)(16,20)(21,37)(22,38)(23,34)(24,36)(26,33)(28,31)(29,30)(32,35)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11,16,24)(10,17,20,36)(12,26,29,40)(13,22,31,37)(14,23,15,32)(18,34,19,35)(21,27,38,28)(25,33,30,39);(9,12,16,29)(10,13,20,31)(11,21,24,38)(14,28,15,27)(17,33,36,39)(18,25,19,30)(22,34,37,35)(23,40,32,26);(2,7)(3,8)(9,13,16,31)(10,12,20,29)(11,22,24,37)(14,30,15,25)(17,26,36,40)(18,27,19,28)(21,34,38,35)(23,39,32,33);(1,4,5,6)(2,3,7,8)(9,14,16,15)(10,18,20,19)(11,23,24,32)(12,27,29,28)(13,30,31,25)(17,34,36,35)(21,26,38,40)(22,39,37,33)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409762);