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 | |
4104 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)(3,8)(4,7)(5,9)(6,10)(11,16)(12,15)(13,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34,39,43)(35,44,47,40)(36,42,46,41)(37,38,45,48);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,13)(15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,39,47)(34,40,43,44)(36,45,46,37)(38,41,48,42);(4,12)(7,15)(11,13)(14,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,36,39,46)(34,41,43,42)(35,45,47,37)(38,40,48,44);(1,4,6,12)(2,7,10,15)(3,11,9,13)(5,14,8,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36)(34,42)(35,45)(37,47)(38,40)(39,46)(41,43)(44,48);(1,3)(2,8)(4,13)(5,10)(6,9)(7,14)(11,12)(15,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,37,39,45)(34,38,43,48)(35,46,47,36)(40,42,44,41);(1,5,6,8)(2,9,10,3)(4,14,12,16)(7,13,15,11)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,38)(34,37)(35,41)(36,40)(39,48)(42,47)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091096);