Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533075 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,15,20,14,18,13,19)(11,21,24,16,25,33,23,31)(12,26,28,17,29,34,27,32)(22,30,38,39,37,40,36,35);(1,2,6,5)(3,8,4,7)(9,11,14,25)(10,16,18,31)(12,22,29,37)(13,24,15,23)(17,30,32,40)(19,33,20,21)(26,39,34,35)(27,38,28,36);(1,3,6,4)(2,7,5,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,32)(19,26)(20,34)(21,35)(23,36)(24,38)(25,37)(31,40)(33,39);(1,4,6,3)(2,7,5,8)(10,18)(16,31)(17,32)(19,20)(21,33)(26,34)(30,40)(35,39);(1,5,6,2)(3,8,4,7)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,f^b=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480602);