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 | |
2118 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)(11,20)(12,24)(13,26)(14,31)(15,19)(16,23)(17,29)(18,28)(21,36)(22,33)(25,34)(27,32)(30,40)(35,39)(37,38);(2,6)(7,8)(9,11,15,23)(10,16,19,20)(12,14,22,28)(13,21,30,37)(17,34,35,32)(18,33,31,24)(25,29,27,39)(26,38,40,36);(1,3,5,4)(2,7,6,8)(9,12,14,23,15,22,28,11)(10,16,18,24,19,20,31,33)(13,27,29,21,30,25,39,37)(17,34,26,36,35,32,40,38);(1,4,5,3)(2,8,6,7)(9,13,28,39,15,30,14,29)(10,17,31,40,19,35,18,26)(11,21,22,27,23,37,12,25)(16,32,33,36,20,34,24,38);(2,6)(7,8)(9,14,15,28)(10,18,19,31)(11,22,23,12)(13,29,30,39)(16,33,20,24)(17,26,35,40)(21,27,37,25)(32,36,34,38)Polycyclic presentation:
2^9|b^2=i,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,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, 7469024);