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 | |
4297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,11,13,6)(4,9,5,8)(12,15,14,16)(17,18)(19,29)(20,26)(21,23)(22,27)(24,28)(25,32)(30,31)(33,34,39,42)(35,40,45,36)(37,38,41,47)(43,44,48,46);(1,3)(2,6)(4,12)(5,14)(7,13)(8,15)(9,16)(10,11)(17,19)(18,25)(20,30)(21,27)(22,26)(23,31)(24,32)(28,29)(33,35,39,45)(34,40,42,36)(37,43,41,48)(38,44,47,46);(1,4,7,5)(2,8,10,9)(3,12,13,14)(6,15,11,16)(17,20,24,21)(18,26,28,23)(19,30,32,27)(22,29,31,25)(33,36,39,40)(34,35,42,45)(37,46,41,44)(38,43,47,48);(1,5,7,4)(2,9,10,8)(3,12,13,14)(6,15,11,16)(17,21,24,20)(18,26,28,23)(19,27,32,30)(22,29,31,25)(33,37)(34,38)(35,43)(36,46)(39,41)(40,44)(42,47)(45,48);(1,3)(2,6)(4,14)(5,12)(7,13)(8,16)(9,15)(10,11)(17,22,24,31)(18,27,28,30)(19,26,32,23)(20,25,21,29)(33,36)(34,35)(37,46)(38,43)(39,40)(41,44)(42,45)(47,48);(1,6)(2,3)(4,15)(5,16)(7,11)(8,12)(9,14)(10,13)(17,23,24,26)(18,21,28,20)(19,31,32,22)(25,27,29,30)(33,38)(34,41)(35,44)(36,43)(37,42)(39,47)(40,48)(45,46)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=gi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440744);