Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 | |
2175 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)(9,10,16,19)(11,17,21,20)(12,14,26,15)(13,18,28,35)(22,32,38,37)(23,33,30,24)(25,29,36,31)(27,34,39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,27)(14,23)(15,30)(16,21)(18,34)(19,20)(22,29)(25,37)(26,33)(28,39)(31,38)(32,36)(35,40);(9,12,14,19,16,26,15,10)(11,20,23,33,21,17,30,24)(13,25,29,35,28,36,31,18)(22,37,39,40,38,32,27,34);(2,6)(5,7)(11,21)(13,28)(17,20)(18,35)(23,30)(24,33)(25,36)(29,31);(3,8)(5,7)(9,13)(10,18)(11,22)(12,25)(14,29)(15,31)(16,28)(17,32)(19,35)(20,37)(21,38)(23,39)(24,34)(26,36)(27,30)(33,40)Polycyclic presentation:
2^9|a^2=i,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,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, 6589855);