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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1769 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)(3,5)(4,6)(7,8)(9,10,15,19,16,11,14,22)(12,23,25,20,26,34,32,17)(13,18,30,36,31,21,29,27)(24,37,39,35,28,38,40,33);(1,2)(3,7)(4,6)(5,8)(9,11,16,10)(12,17,26,20)(13,27,31,36)(14,19,15,22)(18,29,21,30)(23,32,34,25)(24,38,28,37)(33,40,35,39);(1,3,4,8)(2,5,6,7)(9,12,16,26)(10,17,11,20)(13,28,31,24)(14,25,15,32)(18,35,21,33)(19,34,22,23)(27,38,36,37)(29,40,30,39);(3,8)(5,7)(9,13)(10,18)(11,21)(12,24)(14,29)(15,30)(16,31)(17,33)(19,36)(20,35)(22,27)(23,37)(25,39)(26,28)(32,40)(34,38);(2,6)(5,7)(9,14,16,15)(10,19,11,22)(12,25,26,32)(13,29,31,30)(17,34,20,23)(18,36,21,27)(24,39,28,40)(33,38,35,37)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,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, 6970324);