Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | |
1736 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,27)(13,29)(14,19)(15,21)(17,30)(18,32)(20,33)(22,23)(24,36)(25,38)(26,37)(28,34)(31,39)(35,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,28)(13,31)(14,24)(15,26)(17,23)(18,35)(19,36)(20,38)(21,37)(22,30)(25,33)(27,34)(29,39)(32,40);(1,4,5,3)(2,6,7,8)(9,12,15,18)(10,17,21,13)(11,22,26,39)(14,29,33,30)(16,34,37,40)(19,27,20,32)(23,36,31,38)(24,28,25,35);(9,13,15,17)(10,18,21,12)(11,23,26,31)(14,27,33,32)(16,35,37,28)(19,29,20,30)(22,36,39,38)(24,40,25,34);(2,7)(6,8)(9,14,15,33)(10,19,21,20)(11,24,26,25)(12,29,18,30)(13,32,17,27)(16,36,37,38)(22,28,39,35)(23,34,31,40)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422663);