Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021351 |
| 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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1564 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)(17,18,23,40,24,28,22,29)(19,30,39,25,32,43,33,44)(20,34,36,26,37,46,21,27)(31,41,48,42,35,45,38,47);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,19,24,32)(18,25,28,44)(20,35,37,31)(21,38,36,48)(22,39,23,33)(26,45,27,41)(29,43,40,30)(34,47,46,42);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,22,21,24,37,23,36)(18,26,40,46,28,27,29,34)(19,31,39,48,32,35,33,38)(25,41,30,47,44,45,43,42);(4,11)(7,9)(12,16)(14,15)(17,21,22,37,24,36,23,20)(18,27,29,46,28,26,40,34)(19,31,33,38,32,35,39,48)(25,42,30,41,44,47,43,45);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(18,28)(19,32)(26,27)(29,40)(31,35)(33,39)(34,46)(38,48)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=gh,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=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105839);