Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | |
2217 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)(3,5)(4,6)(7,8)(9,10,13,17,14,19,27,18)(11,20,23,32,24,31,22,15)(12,16,28,34,29,26,40,35)(21,33,25,36,38,39,37,30);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,25)(13,22)(14,24)(16,33)(17,31)(18,20)(19,32)(21,28)(23,27)(26,39)(29,37)(30,34)(35,36)(38,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,26)(13,17)(14,19)(16,29)(18,27)(20,23)(21,36)(22,31)(24,32)(25,39)(28,35)(30,38)(33,37)(34,40);(3,8)(5,7)(9,12,27,40,14,29,13,28)(10,16,18,35,19,26,17,34)(11,21,22,37,24,38,23,25)(15,30,31,39,32,36,20,33);(2,6)(5,7)(9,13,14,27)(10,17,19,18)(11,22,24,23)(12,28,29,40)(15,31,32,20)(16,34,26,35)(21,37,38,25)(30,39,36,33)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903607);