Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
2068 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)(3,5)(4,6)(7,8)(9,10,15,21)(11,22,27,33)(12,18,30,38)(13,20,14,19)(16,25,34,26)(17,31,36,29)(23,32,28,35)(24,39,40,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,28)(13,26)(14,25)(15,27)(17,35)(18,37)(19,22)(20,33)(21,34)(23,30)(24,29)(31,40)(32,36)(38,39);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,23)(13,31)(14,29)(15,30)(16,32)(18,20)(19,38)(21,36)(22,37)(24,26)(25,40)(27,28)(33,39)(34,35);(3,8)(5,7)(9,12)(10,18)(11,24)(13,29)(14,31)(15,30)(16,32)(17,20)(19,36)(21,38)(22,39)(23,26)(25,28)(27,40)(33,37)(34,35);(2,6)(5,7)(9,13,15,14)(10,19,21,20)(11,25,27,26)(12,29,30,31)(16,33,34,22)(17,18,36,38)(23,24,28,40)(32,37,35,39)Polycyclic presentation:
2^9|a^2=i,c^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962164);