Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367534 |
| 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 | |
1506 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)(3,12)(4,13)(5,9)(6,10)(7,16)(8,11)(14,15)(18,24)(21,22)(25,37)(26,38)(27,35)(29,34)(31,42)(32,36)(39,47)(40,41)(43,48)(44,45);(1,3,8,14)(2,5,11,4)(6,9,7,13)(10,15,16,12)(17,18,23,27)(19,25,30,41)(20,31,33,38)(21,24,22,35)(26,36,42,32)(28,43,46,47)(29,40,34,37)(39,44,48,45);(1,4)(2,3)(5,8)(6,12)(7,15)(9,10)(11,14)(13,16)(17,19)(18,25)(20,28)(21,34)(22,29)(23,30)(24,37)(26,39)(27,41)(31,43)(32,45)(33,46)(35,40)(36,44)(38,47)(42,48);(1,5)(2,9)(3,10)(4,8)(6,12)(7,15)(11,13)(14,16)(17,20,23,33)(18,26,27,42)(19,28,30,46)(21,32,22,36)(24,38,35,31)(25,39,41,48)(29,44,34,45)(37,47,40,43);(1,6,8,7)(2,10,11,16)(3,13,14,9)(4,15,5,12)(17,21,23,22)(18,24,27,35)(19,29,30,34)(20,32,33,36)(25,40,41,37)(26,38,42,31)(28,44,46,45)(39,43,48,47)Polycyclic presentation:
2^9|b^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274884);