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 | |
4258 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30)(33,34,38,40)(35,43,44,39)(36,45,46,41)(37,42,47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,38,44)(34,39,40,43)(36,37,46,47)(41,42,45,48);(2,6)(3,5)(4,12)(15,16)(19,24)(20,23)(21,29)(25,32)(34,40)(36,46)(37,47)(39,43);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31)(33,36,38,46)(34,41,40,45)(35,37,44,47)(39,42,43,48);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27)(33,37)(34,42)(35,36)(38,47)(39,41)(40,48)(43,45)(44,46);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,34)(35,39)(36,41)(37,42)(38,40)(43,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=h,e^2=gh,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, 8886913);