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 | |
1736 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)(11,22)(12,27)(13,31)(14,19)(15,20)(16,21)(17,26)(18,29)(23,24)(25,38)(28,39)(30,33)(32,40)(34,37)(35,36);(1,3)(2,6)(4,7)(5,8)(9,11,15,26)(10,17,20,22)(12,28,30,36)(13,32,18,24)(14,25,16,34)(19,37,21,38)(23,29,40,31)(27,35,33,39);(9,12,15,30)(10,18,20,13)(11,23,26,40)(14,31,16,29)(17,35,22,39)(19,27,21,33)(24,37,32,38)(25,36,34,28);(2,7)(3,8)(9,13)(10,12)(11,24)(14,27)(15,18)(16,33)(17,36)(19,29)(20,30)(21,31)(22,28)(23,37)(25,39)(26,32)(34,35)(38,40);(1,4,5,6)(2,3,7,8)(9,14,15,16)(10,19,20,21)(11,25,26,34)(12,29,30,31)(13,27,18,33)(17,37,22,38)(23,28,40,36)(24,39,32,35)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=f,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, 7409759);