Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2686 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24)(13,29)(14,20)(15,19)(17,26)(18,22)(21,35)(23,32)(25,31)(27,36)(28,37)(30,33)(34,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,34)(18,29)(19,32)(20,33)(21,27)(24,31)(26,38)(28,39)(35,36)(37,40);(9,12,15,28)(10,17,19,36)(11,21,23,38)(13,27,14,26)(16,31,32,40)(18,35,33,34)(20,37,29,24)(22,39,30,25);(2,6)(3,8)(9,11)(10,18)(12,21)(13,30)(14,22)(15,23)(16,29)(17,35)(19,33)(20,32)(24,31)(25,27)(26,39)(28,38)(34,36)(37,40);(10,19)(16,32)(17,36)(18,33)(20,29)(24,37)(31,40)(34,35)Polycyclic presentation:
2^9|c^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407933);