Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 | |
1908 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)(11,23)(12,27)(13,20)(14,21)(15,18)(16,22)(17,26)(19,24)(25,39)(28,36)(29,35)(30,37)(31,34)(32,38)(33,40);(2,6)(7,8)(9,11,14,26)(10,17,21,23)(12,16,24,15)(13,29,30,28)(18,19,22,27)(20,36,37,35)(25,32,40,31)(33,38,39,34);(10,18)(11,24)(12,26)(15,16)(19,27)(20,34)(21,22)(25,29)(28,40)(31,32)(35,36)(37,38);(1,3,5,4)(2,7,6,8)(9,12)(10,19)(11,16)(13,25)(14,24)(15,26)(17,22)(18,23)(20,33)(21,27)(28,31)(29,32)(30,40)(34,35)(36,38)(37,39);(1,4,5,3)(2,8,6,7)(9,13,14,30)(10,20,21,37)(11,25,26,40)(12,28,24,29)(15,31,16,32)(17,33,23,39)(18,34,22,38)(19,35,27,36)Polycyclic presentation:
2^9|b^2=i,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485114);