Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30)(33,34)(35,41)(36,40)(37,38)(39,42)(43,48)(44,47)(45,46);(2,7)(4,14)(8,11)(10,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(35,42)(39,41)(43,44)(47,48);(2,7)(3,12)(4,14)(5,16)(19,24)(20,23)(21,29)(25,32)(33,35,38,42)(34,39,37,41)(36,44,46,43)(40,48,45,47);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31)(33,36,38,46)(34,40,37,45)(35,43,42,44)(39,47,41,48);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27)(33,35)(34,41)(36,44)(37,39)(38,42)(40,47)(43,46)(45,48);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,37,38,34)(35,39,42,41)(36,45,46,40)(43,47,44,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=i,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900305);