Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,15,20,14,19,13,21)(11,22,26,16,25,38,27,35)(12,28,30,17,32,18,31,36)(23,33,40,39,29,34,24,37);(1,2,5,6)(3,4,8,7)(9,11,14,25)(10,16,19,35)(12,29,32,23)(13,27,15,26)(17,34,36,33)(18,37,28,39)(20,38,21,22)(24,31,40,30);(1,3)(2,7)(4,6)(5,8)(9,12,14,32)(10,17,19,36)(11,23,25,29)(13,30,15,31)(16,33,35,34)(18,20,28,21)(22,39,38,37)(24,27,40,26);(2,6)(3,8)(9,12)(10,18)(11,24)(13,31)(14,32)(15,30)(16,34)(17,21)(19,28)(20,36)(22,39)(23,26)(25,40)(27,29)(33,35)(37,38);(1,4,5,7)(2,3,6,8)(10,19)(11,25)(17,36)(18,28)(20,21)(23,29)(24,40)(26,27)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089204);