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 |
| prime radical | #61054 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1905 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)(9,10,14,18,15,12,26,19)(11,20,24,21,25,32,23,16)(13,17,29,34,30,27,37,35)(22,33,28,36,39,40,38,31);(1,2)(3,7)(4,6)(5,8)(9,11,15,25)(10,16,12,21)(13,28,30,38)(14,23,26,24)(17,33,27,40)(18,32,19,20)(22,37,39,29)(31,35,36,34);(1,3,4,8)(2,5,6,7)(9,12)(10,15)(11,21)(13,17)(14,19)(16,25)(18,26)(20,23)(22,31)(24,32)(27,30)(28,33)(29,34)(35,37)(36,39)(38,40);(3,8)(5,7)(9,13,26,37,15,30,14,29)(10,17,19,35,12,27,18,34)(11,22,23,38,25,39,24,28)(16,31,32,40,21,36,20,33);(2,6)(5,7)(9,14,15,26)(10,18,12,19)(11,23,25,24)(13,29,30,37)(16,32,21,20)(17,34,27,35)(22,38,39,28)(31,40,36,33)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,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, 6963128);