Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1615 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)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,24,30,23,28,22,29)(19,31,36,25,34,47,35,44)(20,37,40,26,33,46,39,43)(21,27,32,42,41,48,38,45);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,23,34)(18,25,28,44)(20,38,33,32)(21,40,41,39)(22,36,24,35)(26,45,43,42)(27,46,48,37)(29,47,30,31);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20,23,33)(18,26,28,43)(19,32,34,38)(21,36,41,35)(22,39,24,40)(25,42,44,45)(27,47,48,31)(29,37,30,46);(4,12)(7,16)(9,11)(13,15)(17,21,24,32,23,41,22,38)(18,27,30,42,28,48,29,45)(19,33,36,39,34,20,35,40)(25,43,47,37,44,26,31,46);(2,8)(3,10)(7,16)(9,11)(18,28)(19,34)(21,41)(26,43)(29,30)(32,38)(35,36)(37,46)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7012009);