Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 | |
1812 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,13)(14,19)(15,20)(16,18)(17,24)(22,26)(23,37)(25,33)(27,28)(29,40)(30,36)(31,34)(32,39)(35,38);(2,5)(3,7)(9,11)(10,17)(12,26)(13,25)(14,29)(15,24)(16,22)(18,33)(19,35)(20,21)(23,31)(27,37)(28,39)(30,38)(32,34)(36,40);(9,12,13,10,15,18,16,20)(11,17,25,33,24,21,22,26)(14,27,28,19,30,34,31,36)(23,32,29,40,39,37,38,35);(2,5)(6,8)(9,13,15,16)(10,18,20,12)(11,22,24,25)(14,28,30,31)(17,26,21,33)(19,34,36,27)(23,38,39,29)(32,35,37,40);(1,3,4,7)(2,6,5,8)(9,14,15,30)(10,19,20,36)(11,23,24,39)(12,27,18,34)(13,28,16,31)(17,32,21,37)(22,38,25,29)(26,35,33,40)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592706);