Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 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)(3,4)(5,7)(6,8)(9,10,13,18,14,19,29,36)(11,15,21,17,23,32,22,31)(12,24,26,34,27,35,28,16)(20,37,38,40,39,33,25,30);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,25)(13,21)(14,23)(16,33)(17,18)(19,32)(20,26)(22,29)(24,30)(27,38)(28,39)(31,36)(34,37)(35,40);(9,12,14,27)(10,16,19,34)(11,20,23,39)(13,28,29,26)(15,30,32,40)(17,33,31,37)(18,35,36,24)(21,25,22,38);(2,7)(3,8)(9,11)(10,17)(12,20)(13,22)(14,23)(15,18)(16,33)(19,31)(21,29)(24,40)(25,26)(27,39)(28,38)(30,35)(32,36)(34,37);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7411145);