Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,23,30,24,19,22,27)(20,31,35,25,33,44,34,28)(21,26,39,36,40,29,38,43)(32,45,47,41,37,48,46,42);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,19,24,18)(20,25,33,28)(21,36,40,43)(22,27,23,30)(26,38,29,39)(31,34,44,35)(32,45,37,48)(41,47,42,46);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,23,35,24,33,22,34)(18,25,27,31,19,28,30,44)(21,37,39,46,40,32,38,47)(26,42,43,48,29,41,36,45);(4,11)(7,9)(12,16)(14,15)(17,21,22,38,24,40,23,39)(18,26,27,43,19,29,30,36)(20,32,34,46,33,37,35,47)(25,41,31,45,28,42,44,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(18,19)(20,33)(26,29)(27,30)(32,37)(34,35)(36,43)(46,47)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=ghi,d^2=f,e^2=hi,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,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105844);