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 | |
4107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,8,14,10)(12,15,13,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(2,3)(7,9)(8,10)(15,16)(17,19,22,26)(18,23,24,21)(20,28,29,27)(25,31,32,30)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(4,11)(7,9)(12,13)(15,16)(18,24)(19,26)(20,29)(30,31)(34,40)(36,44)(39,46)(41,47);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,19)(18,21)(20,27)(22,26)(23,24)(25,31)(28,29)(30,32)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,4,6,11)(2,7,3,9)(5,13,14,12)(8,16,10,15)(17,20,22,29)(18,25,24,32)(19,27,26,28)(21,31,23,30)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,21)(18,19)(20,30)(22,23)(24,26)(25,28)(27,32)(29,31)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962745);