Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500002 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,15)(12,22,23,34)(13,18)(14,17)(16,24,31,26)(19,35,36,39)(20,30)(21,29)(25,33)(27,32)(28,37,40,38);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,29)(12,19,23,36)(13,25,27,20)(16,28,31,40)(18,33,32,30)(22,39,34,35)(24,38,26,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,31)(18,34)(20,38)(21,36)(22,32)(24,27)(25,37)(29,40)(30,39)(33,35);(3,8)(5,7)(10,17)(15,29)(16,31)(18,32)(22,34)(28,40)(30,33)(35,39);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317675 |
| maximal normal subgroup of | #72317675 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407199);