Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
1805 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)(3,5)(4,6)(7,8)(9,10,15,20,16,21,14,19)(11,22,24,17,12,25,27,33)(13,18,30,36,31,37,29,35)(23,38,39,32,26,34,28,40);(1,2)(3,7)(4,6)(5,8)(9,11,16,12)(10,17,21,33)(13,28,31,39)(14,27,15,24)(18,34,37,38)(19,22,20,25)(23,29,26,30)(32,36,40,35);(1,3,4,8)(2,5,6,7)(9,12,16,11)(10,17,21,33)(13,26,31,23)(14,27,15,24)(18,32,37,40)(19,25,20,22)(28,30,39,29)(34,36,38,35);(3,8)(5,7)(9,13,16,31)(10,18,21,37)(11,23,12,26)(14,29,15,30)(17,32,33,40)(19,35,20,36)(22,38,25,34)(24,39,27,28);(2,6)(5,7)(9,14,16,15)(10,19,21,20)(11,24,12,27)(13,29,31,30)(17,25,33,22)(18,35,37,36)(23,39,26,28)(32,34,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970314);