Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | |
2094 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)(3,7)(4,8)(5,6)(9,10)(11,21)(12,26)(13,16)(14,20)(15,19)(17,25)(18,31)(22,38)(23,33)(24,32)(27,37)(28,30)(29,36)(34,40)(35,39);(2,6)(7,8)(9,11)(10,17)(12,27)(13,32)(14,24)(15,25)(16,33)(18,35)(19,21)(20,23)(22,31)(26,34)(28,37)(29,39)(30,40)(36,38);(1,3,5,4)(2,7,6,8)(9,12,16,31,15,30,14,29)(10,18,20,26,19,36,13,28)(11,22,24,27,25,39,33,40)(17,34,32,35,21,37,23,38);(1,4,5,3)(2,8,6,7)(9,13,14,19,15,20,16,10)(11,23,33,21,25,32,24,17)(12,28,29,36,30,26,31,18)(22,38,40,37,39,35,27,34);(2,6)(7,8)(9,14,15,16)(10,13,19,20)(11,24,25,33)(12,29,30,31)(17,32,21,23)(18,28,36,26)(22,27,39,40)(34,35,37,38)Polycyclic presentation:
2^9|c^2=fgi,d^2=fg,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,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468968);