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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,15,21)(11,22,25,17)(12,26,29,36)(13,31,16,19)(14,20,23,38)(18,28,39,30)(24,35,33,37)(27,40,32,34);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,27)(13,24)(14,30)(15,25)(16,33)(18,38)(19,35)(20,39)(21,22)(23,28)(26,34)(29,32)(31,37)(36,40);(9,12,15,29)(10,18,21,39)(11,23,25,14)(13,28,16,30)(17,34,22,40)(19,36,31,26)(20,35,38,37)(24,32,33,27);(1,3,5,4)(2,6,7,8)(9,13,15,16)(10,19,21,31)(11,24,25,33)(12,28,29,30)(14,27,23,32)(17,35,22,37)(18,36,39,26)(20,34,38,40);(1,4,5,3)(2,6,7,8)(9,14)(10,20)(11,12)(13,32)(15,23)(16,27)(17,36)(18,35)(19,40)(21,38)(22,26)(24,30)(25,29)(28,33)(31,34)(37,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,e^2=fi,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, 6583717);