Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,6,8,5,9,4,13)(3,7,11,16,12,10,14,15)(17,18)(19,30,34,43)(20,35)(21,27)(22,28)(23,40)(24,29)(25,33,44,39)(26,36)(31,41,32,45)(37,46)(38,48,47,42);(1,2,4,13,5,9,6,8)(3,10,14,16,12,7,11,15)(17,19,23,33,24,34,22,39)(18,25,40,30,29,44,28,43)(20,31,36,47,37,32,21,38)(26,41,46,42,27,45,35,48);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,10)(13,15)(17,20,22,21,24,37,23,36)(18,26,40,46,29,27,28,35)(19,31,39,38,34,32,33,47)(25,41,30,42,44,45,43,48);(1,4,5,6)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,22,37,24,36,23,20)(18,27,28,46,29,26,40,35)(19,32,33,38,34,31,39,47)(25,42,30,45,44,48,43,41)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138714);