Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1943 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)(3,5)(4,6)(7,8)(9,10)(11,21,25,34)(12,26,29,36)(13,18)(14,19)(15,20)(16,23,35,24)(17,28,33,27)(22,37)(30,38)(31,39)(32,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,22)(13,27)(14,24)(15,25)(17,32)(18,26)(19,34)(20,35)(21,39)(23,31)(28,30)(29,40)(33,37)(36,38);(1,3)(2,5)(4,8)(6,7)(9,12,15,29)(10,17,20,33)(11,22,25,40)(13,23,30,24)(14,27,31,28)(16,32,35,37)(18,34,38,21)(19,36,39,26);(3,8)(5,7)(9,13,14,22,15,30,31,40)(10,18,19,37,20,38,39,32)(11,12,23,27,25,29,24,28)(16,33,21,26,35,17,34,36);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914399);