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 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,13,15)(10,12,16,14)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34,39,40)(35,42,36,41)(37,43,47,46)(38,45,48,44);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(34,40)(37,47)(41,42)(44,45);(5,13)(8,15)(10,16)(12,14)(19,25)(23,27)(26,31)(30,32)(33,35,39,36)(34,41,40,42)(37,45,47,44)(38,46,48,43);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36,39,35)(34,42,40,41)(37,45,47,44)(38,46,48,43);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,37)(34,43)(35,45)(36,44)(38,41)(39,47)(40,46)(42,48);(1,5,6,13)(2,8,9,15)(3,10,11,16)(4,12,7,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,38,39,48)(34,44,40,45)(35,43,36,46)(37,41,47,42)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806806);