Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 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)(3,4)(5,7)(6,8)(9,10)(11,16)(12,24)(13,29)(14,19)(15,20)(17,27)(18,22)(21,35)(23,33)(25,31)(26,37)(28,36)(30,32)(34,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,34)(18,29)(19,32)(20,33)(21,28)(24,31)(26,39)(27,38)(35,36)(37,40);(9,12,15,26)(10,17,20,36)(11,21,23,38)(13,28,14,27)(16,31,33,40)(18,35,32,34)(19,37,29,24)(22,39,30,25);(2,7)(3,8)(9,11)(10,18)(12,21)(13,30)(14,22)(15,23)(16,29)(17,35)(19,33)(20,32)(24,31)(25,28)(26,38)(27,39)(34,36)(37,40);(1,4,5,6)(2,3,7,8)(12,26)(17,36)(21,38)(24,37)(25,39)(27,28)(31,40)(34,35)Polycyclic presentation:
2^9|c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7409751);