Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| 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 | |
2090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,21)(12,13)(14,34)(15,20)(16,25)(17,29)(18,32)(19,28)(22,39)(23,27)(24,36)(26,37)(30,33)(31,40)(35,38);(2,6)(7,8)(9,11,15,25)(10,16,20,21)(12,27,30,40)(13,31,33,23)(14,24,28,26)(17,38,32,39)(18,35,29,22)(19,36,34,37);(1,3,5,4)(2,7,6,8)(9,12,28,18,15,30,14,29)(10,17,34,33,20,32,19,13)(11,22,24,40,25,35,26,27)(16,31,36,39,21,23,37,38);(1,4,5,3)(2,8,6,7)(9,13,14,32,15,33,28,17)(10,18,19,12,20,29,34,30)(11,23,26,39,25,31,24,38)(16,35,37,40,21,22,36,27);(2,6)(7,8)(9,14,15,28)(10,19,20,34)(11,24,25,26)(12,29,30,18)(13,32,33,17)(16,36,21,37)(22,40,35,27)(23,38,31,39)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fgi,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,d^c=di,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469006);