Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535696 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61318 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,15,19,14,20,13,18)(11,21,23,31,25,34,24,16)(12,26,27,33,28,17,29,32)(22,36,38,40,39,35,37,30);(1,3,6,4)(2,7,5,8)(9,11,13,24,14,25,15,23)(10,16,19,21,20,31,18,34)(12,22,27,38,28,39,29,37)(17,30,33,35,26,40,32,36);(1,2,6,5)(3,8,4,7)(9,12,13,27,14,28,15,29)(10,17,19,33,20,26,18,32)(11,22,24,38,25,39,23,37)(16,30,21,35,31,40,34,36);(1,4,6,3)(2,7,5,8)(9,13,14,15)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,31,34)(17,32,26,33)(22,37,39,38)(30,35,40,36);(1,5,6,2)(3,8,4,7)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=f,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,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, 6461862);