Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6363286 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1604 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,24,30)(19,31,35,46)(20,26,21,27)(22,29,23,28)(25,39,42,34)(32,47,36,41)(33,43,48,40)(37,45,38,44);(1,3,5,11)(2,6,8,14)(4,10,12,16)(7,13,9,15)(17,19)(18,25)(20,36)(21,32)(22,34)(23,39)(24,35)(26,43)(27,40)(28,31)(29,46)(30,42)(33,38)(37,48)(41,45)(44,47);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20,24,21)(18,26,30,27)(19,32,35,36)(22,38,23,37)(25,40,42,43)(28,45,29,44)(31,47,46,41)(33,34,48,39);(17,21,24,20)(18,27,30,26)(19,33,35,48)(22,38,23,37)(25,41,42,47)(28,45,29,44)(31,43,46,40)(32,34,36,39);(2,8)(6,14)(7,9)(13,15)(17,22,24,23)(18,28,30,29)(19,34,35,39)(20,37,21,38)(25,31,42,46)(26,44,27,45)(32,33,36,48)(40,41,43,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6270515);