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 | |
4068 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)(18,23)(19,25)(21,29)(28,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(2,6)(7,9)(8,12)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(2,6)(3,10)(8,12)(11,16)(19,25)(20,24)(27,32)(28,31)(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)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28)(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)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(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)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=gh,e^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, 8034696);