Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
2670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,22)(12,26)(13,21)(14,19)(15,20)(16,24)(17,30)(18,32)(23,35)(25,40)(27,34)(28,38)(29,36)(31,39)(33,37);(1,3)(2,5)(4,8)(6,7)(9,11,15,24)(10,16,20,22)(12,27,29,31)(13,23,32,25)(14,28,33,30)(17,37,38,19)(18,35,21,40)(26,39,36,34);(9,12)(10,17)(11,14)(13,28)(15,29)(16,34)(18,36)(19,35)(20,38)(21,26)(22,39)(23,27)(24,33)(25,31)(30,32)(37,40);(2,6)(5,7)(9,13,15,32)(10,18,20,21)(11,23,24,25)(12,28,29,30)(14,27,33,31)(16,35,22,40)(17,36,38,26)(19,39,37,34);(3,8)(5,7)(9,14)(10,19)(11,12)(13,31)(15,33)(16,36)(17,35)(18,34)(20,37)(21,39)(22,26)(23,30)(24,29)(25,28)(27,32)(38,40)Polycyclic presentation:
2^9|b^2=i,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575170);