Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2084 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)(4,6)(5,8)(9,10)(11,21)(12,25)(13,20)(14,16)(15,19)(17,22)(18,26)(23,33)(24,32)(27,34)(28,30)(29,31)(35,37)(36,40)(38,39);(1,3,5,4)(2,6,8,7)(10,16)(11,22)(13,15)(17,32)(21,30)(23,37)(24,28)(25,34)(26,29)(33,40)(35,39)(36,38);(9,11,14,24)(10,17,16,30)(12,23,27,38)(13,22,15,28)(18,33,31,39)(19,21,20,32)(25,35,34,40)(26,36,29,37);(1,3,5,4)(2,7,8,6)(9,12)(10,18)(11,23)(13,26)(14,27)(15,29)(16,31)(17,33)(19,25)(20,34)(21,35)(22,36)(24,38)(28,37)(30,39)(32,40);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,19,16,20)(11,22,24,28)(12,26,27,29)(17,21,30,32)(18,25,31,34)(23,36,38,37)(33,35,39,40)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=fi,e^2=f,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,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429705);