Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574086 |
| 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 | |
2004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,17)(13,19)(14,20)(16,29)(18,28)(21,39)(22,33)(23,25)(24,38)(26,34)(27,32)(30,36)(31,37)(35,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,27)(13,24)(14,26)(16,35)(17,32)(18,23)(19,38)(20,34)(21,30)(22,31)(25,28)(29,40)(33,37)(36,39);(9,12,28,30,14,31,13,29)(10,16,19,37,20,36,18,17)(11,21,25,22,26,40,24,27)(15,32,38,35,34,33,23,39);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,29)(14,31)(15,33)(16,19)(18,36)(20,37)(21,24)(23,35)(25,40)(26,27)(28,30)(32,34)(38,39);(9,10)(11,23)(12,16)(13,18)(14,20)(15,25)(17,29)(19,28)(21,39)(22,32)(24,34)(26,38)(27,33)(30,37)(31,36)(35,40)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491548);