Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366206 |
| 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 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,9,12,15)(4,13,14,8)(6,10,16,11)(17,18,23,26)(19,24,30,38)(20,31,33,42)(21,27,22,35)(25,36,41,32)(28,43,46,48)(29,39,34,40)(37,44,47,45);(1,3)(2,6)(4,10)(5,12)(7,16)(8,15)(9,13)(11,14)(17,19,23,30)(18,24,26,38)(20,28,33,46)(21,34,22,29)(25,37,41,47)(27,40,35,39)(31,43,42,48)(32,45,36,44);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,16)(13,14)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,41)(27,42)(29,44)(30,46)(31,35)(34,45)(38,47)(39,43)(40,48);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(18,26)(24,38)(25,41)(27,35)(31,42)(37,47)(39,40)(43,48);(2,7)(6,16)(8,13)(9,15)(17,21,23,22)(18,27,26,35)(19,29,30,34)(20,32,33,36)(24,39,38,40)(25,42,41,31)(28,44,46,45)(37,43,47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273551);