Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
1952 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)(3,6)(4,8)(5,7)(9,10,15,21)(11,16,26,36)(12,27,29,39)(13,30,32,18)(14,20,33,19)(17,28,34,23)(22,40,31,37)(24,38,25,35);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,13)(14,24)(15,26)(17,37)(18,39)(19,35)(20,38)(21,36)(22,28)(23,31)(25,33)(27,30)(29,32)(34,40);(9,12,14,28,15,29,33,23)(10,17,20,27,21,34,19,39)(11,22,25,13,26,31,24,32)(16,30,35,37,36,18,38,40);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,7,3,8)(9,13,14,31,15,32,33,22)(10,18,20,40,21,30,19,37)(11,23,25,12,26,28,24,29)(16,34,35,39,36,17,38,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718701);