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 | |
4229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,29)(27,32)(30,31)(33,34,38,41)(35,36,39,43)(37,40,46,48)(42,44,45,47);(2,6)(7,9)(8,12)(15,16)(18,23)(20,29)(21,24)(27,30)(33,35)(34,36)(37,45)(38,39)(40,47)(41,43)(42,46)(44,48);(3,10)(7,9)(11,13)(15,16)(18,23)(19,26)(25,28)(27,30)(35,39)(36,43)(42,45)(44,47);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,19,22,26)(18,24,23,21)(20,30,29,27)(25,31,28,32)(33,36,38,43)(34,39,41,35)(37,44,46,47)(40,45,48,42);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(17,20,22,29)(18,25,23,28)(19,27,26,30)(21,31,24,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21,22,24)(18,26,23,19)(20,31,29,32)(25,30,28,27)(33,37)(34,40)(35,42)(36,44)(38,46)(39,45)(41,48)(43,47)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8747487);