Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,6,10,7,4,5,9)(3,11,13,12,15,16,14,8)(17,18)(19,29)(20,33,36,42)(21,26)(22,27)(23,28)(24,31)(25,34,41,35)(30,44,47,38)(32,39)(37,40)(43,46,48,45);(1,3,7,15)(2,8,4,12)(5,13,6,14)(9,11,10,16)(17,19,22,37,23,32,21,31)(18,24,26,39,28,40,27,29)(20,30,35,46,36,47,34,45)(25,38,33,48,41,44,42,43);(1,4)(2,7)(3,12)(5,10)(6,9)(8,15)(11,14)(13,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,33)(27,42)(28,41)(29,43)(31,45)(32,47)(37,46)(39,48)(40,44);(1,5,7,6)(2,9,4,10)(3,13,15,14)(8,11,12,16)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,34,36,35)(24,39,40,29)(25,33,41,42)(30,45,47,46)(38,48,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106841);