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 | |
1898 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)(9,10,16,19)(11,22,17,34)(12,25,27,24)(13,28,32,18)(14,21,15,20)(23,36,35,39)(26,40,30,29)(31,37,33,38);(2,5)(3,7)(9,11)(10,12)(13,29)(14,25)(15,24)(16,17)(18,36)(19,27)(20,34)(21,22)(23,31)(26,37)(28,39)(30,38)(32,40)(33,35);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,19)(13,30)(14,22)(15,34)(16,27)(18,23)(20,25)(21,24)(26,32)(28,35)(29,37)(31,39)(33,36)(38,40);(2,5)(6,8)(9,13)(10,18)(11,23)(12,26)(14,33)(15,31)(16,32)(17,35)(19,28)(20,38)(21,37)(22,39)(24,40)(25,29)(27,30)(34,36);(1,3,4,7)(2,6,5,8)(10,19)(11,17)(18,28)(20,21)(23,35)(24,25)(29,40)(37,38)Polycyclic presentation:
2^9|a^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,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, 7050020);