Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18,22,21)(19,28,31,25)(20,33,35,44)(23,38,24,27)(26,40,36,37)(29,46,47,43)(30,39,42,32)(34,41,48,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,25)(20,34)(21,28)(22,31)(23,32)(24,39)(26,43)(27,30)(29,37)(33,45)(35,48)(36,46)(38,42)(40,47)(41,44);(17,20,23,37,22,35,24,40)(18,26,27,33,21,36,38,44)(19,29,32,48,31,47,39,34)(25,41,30,43,28,45,42,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,21)(20,35)(25,28)(27,38)(29,47)(30,42)(34,48)(37,40);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21,22,18)(19,30,31,42)(20,36,35,26)(23,38,24,27)(25,32,28,39)(29,43,47,46)(33,37,44,40)(34,41,48,45)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593793);