Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814799 |
| 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 | |
1805 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,18)(13,19)(14,17)(15,20)(21,32)(22,30)(23,31)(24,29)(25,34)(26,35)(27,33)(28,36)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,26)(14,21)(15,24)(17,32)(18,30)(19,35)(20,29)(23,27)(25,38)(28,37)(31,33)(34,39)(36,40);(10,17,20,18)(11,21,24,22)(16,29)(19,33,36,34)(23,37,38,26)(30,32)(31,39)(35,40);(2,6)(5,7)(9,12,15,14)(10,18,20,17)(11,22,24,21)(13,25,28,27)(16,30,29,32)(19,34,36,33)(23,26,38,37)(31,35,39,40);(3,8)(5,7)(9,13,14,27,15,28,12,25)(10,19,17,33,20,36,18,34)(11,23,22,26,24,38,21,37)(16,31,30,35,29,39,32,40)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722000);