Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | C2×#41 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18)(19,31,36,47)(20,37,33,46)(21,27)(22,29)(23,28)(24,30)(25,35,44,34)(26,40,43,39)(32,42)(38,45)(41,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,25)(20,38)(21,39)(22,35)(23,34)(24,36)(26,45)(27,46)(28,47)(29,31)(30,44)(32,33)(37,48)(40,41)(42,43);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,32)(21,34)(22,39)(23,40)(24,33)(25,42)(27,47)(28,37)(29,46)(30,43)(31,48)(35,41)(36,38)(44,45);(3,10)(6,9)(11,16)(14,15)(17,21,22,38,24,41,23,32)(18,27,29,45,30,48,28,42)(19,33,34,40,36,20,35,39)(25,43,47,37,44,26,31,46);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,39,33,40)(21,38,41,32)(25,31,44,47)(26,37,43,46)(27,42,48,45)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090859);