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 | |
1930 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)(10,16,19,27)(11,20,22,28)(18,23,33,37)(21,34,36,38);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,23)(13,27)(14,16)(15,19)(18,24)(20,31)(21,35)(22,32)(25,33)(26,37)(28,29)(30,34)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,22)(10,17,19,32)(12,21,26,36)(13,20,14,28)(16,29,27,31)(18,30,33,40)(23,35,37,39)(24,34,25,38);(3,8)(5,7)(9,12,14,25,15,26,13,24)(10,18,27,37,19,33,16,23)(11,21,28,38,22,36,20,34)(17,30,31,39,32,40,29,35);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=fi,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, 6914393);