Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364762 |
| 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 | |
1542 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)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(17,18,24,29)(19,30,34,45)(20,21,26,38)(22,28,23,27)(25,40,43,33)(31,46,44,32)(35,41,39,37)(36,47,48,42);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19)(18,25)(20,31)(21,36)(22,33)(23,40)(24,34)(26,44)(27,30)(28,45)(29,43)(32,37)(35,47)(38,48)(39,42)(41,46);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20,24,26)(18,21,29,38)(19,31,34,44)(22,39,23,35)(25,36,43,48)(27,41,28,37)(30,46,45,32)(33,42,40,47);(17,21)(18,26)(19,32)(20,29)(22,37)(23,41)(24,38)(25,42)(27,39)(28,35)(30,31)(33,48)(34,46)(36,40)(43,47)(44,45);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(17,22,24,23)(18,27,29,28)(19,33,34,40)(20,35,26,39)(21,37,38,41)(25,30,43,45)(31,47,44,42)(32,48,46,36)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,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=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272085);