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 | |
2069 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)(11,16)(12,25)(13,28)(14,19)(15,21)(17,24)(18,30)(20,26)(22,37)(23,33)(27,32)(29,34)(31,35)(36,40)(38,39);(2,6)(7,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,21)(18,34)(19,33)(20,25)(27,40)(28,37)(29,30)(31,38)(32,36)(35,39);(1,3,5,4)(2,7,6,8)(9,12,26,11,15,23,14,24)(10,17,19,33,21,16,20,25)(13,29,40,38,31,27,30,22)(18,32,35,39,36,34,28,37);(1,4,5,3)(2,8,6,7)(9,13,26,40,15,31,14,30)(10,18,19,35,21,36,20,28)(11,22,23,29,24,38,12,27)(16,32,25,39,17,34,33,37);(2,6)(7,8)(9,14,15,26)(10,19,21,20)(11,23,24,12)(13,30,31,40)(16,33,17,25)(18,35,36,28)(22,29,38,27)(32,37,34,39)Polycyclic presentation:
2^9|c^2=fgi,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,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, 7469018);