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 | |
4150 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,23,21)(19,20,27,24)(22,29,30,25)(26,31,32,28)(33,34,39,43)(35,37,44,40)(36,41,46,48)(38,42,47,45);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28)(33,35)(34,40)(36,38)(37,43)(39,44)(41,45)(42,48)(46,47);(2,6)(3,10)(4,13)(14,15)(18,21)(20,24)(22,30)(26,32)(33,36,39,46)(34,41,43,48)(35,38,44,47)(37,42,40,45);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31)(33,37,39,40)(34,35,43,44)(36,45,46,42)(38,41,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,23,18)(19,24,27,20)(22,29,30,25)(26,31,32,28)(33,36)(34,41)(35,38)(37,42)(39,46)(40,45)(43,48)(44,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,30)(18,25,21,29)(19,26,27,32)(20,28,24,31)(33,38)(34,42)(35,36)(37,41)(39,47)(40,48)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=ghi,e^2=hi,f^2=h,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, 8924857);