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 | |
4159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18)(19,24)(20,26)(21,31)(22,32)(23,25)(27,30)(28,29)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(2,6)(7,9)(8,12)(14,15)(17,19)(18,24)(20,23)(21,27)(22,28)(25,26)(29,32)(30,31)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(2,6)(3,10)(8,12)(11,16)(18,25)(19,20)(21,32)(28,30)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,26,25,24)(21,29,32,27)(22,30,31,28)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034567);