Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
2074 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)(12,26)(13,28)(14,19)(15,30)(16,24)(17,18)(20,29)(22,39)(23,27)(25,34)(31,38)(32,40)(33,35)(36,37);(2,6)(7,8)(9,11,14,24)(10,16,19,21)(12,27,28,40)(13,23,26,32)(15,31,29,25)(17,35,36,39)(18,22,37,33)(20,38,30,34);(1,3,5,4)(2,7,6,8)(9,12,15,18,14,28,29,37)(10,17,20,26,19,36,30,13)(11,22,25,27,24,33,31,40)(16,32,34,35,21,23,38,39);(1,4,5,3)(2,8,6,7)(9,13,29,36,14,26,15,17)(10,18,30,12,19,37,20,28)(11,23,31,35,24,32,25,39)(16,33,38,27,21,22,34,40);(2,6)(7,8)(10,19)(12,28)(16,21)(18,37)(20,30)(22,33)(27,40)(34,38)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,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,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469023);