Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 |
| prime radical | #60697 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,23,29)(19,31,35,25)(20,37,40,46)(21,30,24,27)(22,28,38,48)(26,34,45,39)(32,44,41,47)(33,42,36,43);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,23,35)(18,25,29,31)(20,38,40,22)(21,33,24,36)(26,44,45,47)(27,42,30,43)(28,46,48,37)(32,39,41,34);(17,20,21,39,23,40,24,34)(18,26,30,46,29,45,27,37)(19,32,36,22,35,41,33,38)(25,28,42,47,31,48,43,44);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,23,24)(18,27,29,30)(19,33,35,36)(20,39,40,34)(22,32,38,41)(25,42,31,43)(26,37,45,46)(28,47,48,44);(3,11)(6,9)(10,16)(14,15)(17,22,21,41,23,38,24,32)(18,28,30,47,29,48,27,44)(19,34,36,20,35,39,33,40)(25,26,42,46,31,45,43,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=hi,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,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716312);