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 | |
2065 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)(9,10,16,22)(11,23,28,35)(12,29,13,19)(14,21,15,20)(17,26,36,27)(18,32,37,31)(24,33,30,34)(25,39,40,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,30)(13,24)(14,27)(15,26)(16,28)(18,34)(19,38)(20,23)(21,35)(22,36)(25,32)(29,39)(31,40)(33,37);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,24)(13,16)(14,32)(15,31)(17,33)(19,20)(21,29)(22,37)(23,39)(25,26)(27,40)(28,30)(34,36)(35,38);(3,8)(5,7)(9,13,16,12)(10,19,22,29)(11,25,28,40)(14,32,15,31)(17,34,36,33)(18,21,37,20)(23,39,35,38)(24,27,30,26);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,31,13,32)(17,35,36,23)(18,29,37,19)(24,40,30,25)(33,39,34,38)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=i,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=dh,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, 6962134);