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 | |
4055 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)(3,11)(4,13)(5,10)(6,8)(7,14)(9,15)(12,16)(33,34,39,40)(35,37,43,46)(36,41,45,47)(38,42,44,48);(1,3,6,14)(2,7,8,11)(4,10,15,12)(5,13,16,9)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(33,35,39,43)(34,37,40,46)(36,44,45,38)(41,48,47,42);(2,8)(5,16)(7,11)(9,13)(18,23)(21,30)(24,27)(28,32)(34,40)(35,43)(38,44)(41,47);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,36)(34,41)(35,38)(37,42)(39,45)(40,47)(43,44)(46,48);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,37)(34,35)(36,42)(38,41)(39,46)(40,43)(44,47)(45,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,38)(34,42)(35,36)(37,41)(39,44)(40,48)(43,45)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8135452);