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 | |
1858 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)(12,22)(13,27)(14,19)(15,20)(17,32)(18,34)(21,28)(23,26)(24,33)(25,39)(29,31)(30,37)(35,38)(36,40);(2,6)(7,8)(9,11,15,24)(10,16,20,33)(12,18,26,19)(13,21,32,38)(14,22,34,23)(17,35,27,28)(25,37,40,29)(30,36,31,39);(9,12,15,26)(10,11,20,24)(13,28,32,35)(14,33,34,16)(17,36,27,39)(18,23,19,22)(21,29,38,37)(25,30,40,31);(1,3,5,4)(2,7,6,8)(9,13,15,32)(10,17,20,27)(11,21,24,38)(12,25,26,40)(14,30,34,31)(16,35,33,28)(18,37,19,29)(22,36,23,39);(1,4,5,3)(2,8,6,7)(9,10)(11,12)(13,29)(14,18)(15,20)(16,22)(17,30)(19,34)(21,35)(23,33)(24,26)(25,39)(27,31)(28,38)(32,37)(36,40)Polycyclic presentation:
2^9|b^2=i,c^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, 6485117);