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 | |
1911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,19)(11,20,25,34)(12,26,29,17)(13,30,31,18)(15,24,35,23)(16,27,36,28)(21,33,40,32)(22,37,39,38);(2,5)(3,7)(9,11)(10,15)(12,21)(13,24)(14,25)(16,32)(17,37)(18,34)(19,35)(20,30)(22,27)(23,31)(26,38)(28,39)(29,40)(33,36);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,21)(13,27)(14,29)(15,32)(17,18)(19,36)(20,38)(22,24)(23,39)(25,40)(26,30)(28,31)(33,35)(34,37);(2,5)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,33)(16,18)(19,26)(20,38)(21,24)(23,40)(25,39)(27,31)(30,36)(32,35)(34,37);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,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, 7050012);