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 | |
2073 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)(11,20)(12,24)(13,27)(14,30)(15,19)(16,23)(17,28)(18,25)(21,35)(22,32)(26,33)(29,40)(31,36)(34,39)(37,38);(2,6)(7,8)(9,11)(10,16)(12,25)(13,21)(14,22)(15,23)(17,33)(18,32)(19,20)(24,30)(26,39)(27,38)(28,36)(29,37)(31,34)(35,40);(1,3,5,4)(2,7,6,8)(9,12,25,11,15,22,14,23)(10,16,30,32,19,20,18,24)(13,26,39,21,29,36,28,37)(17,31,40,35,34,33,27,38);(1,4,5,3)(2,8,6,7)(9,13,14,28,15,29,25,39)(10,17,18,27,19,34,30,40)(11,21,12,26,23,37,22,36)(16,31,24,38,20,33,32,35);(2,6)(7,8)(9,14,15,25)(10,18,19,30)(11,22,23,12)(13,28,29,39)(16,32,20,24)(17,27,34,40)(21,36,37,26)(31,35,33,38)Polycyclic presentation:
2^9|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,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, 7468982);