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 | |
4077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,5)(4,8,12,16)(11,15,13,14)(17,18,22,23)(19,27,30,24)(20,25,31,29)(21,28,32,26)(33,34,39,43)(35,40,38,48)(36,37,42,41)(44,46,45,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(18,23)(20,31)(24,27)(26,28)(33,35,39,38)(34,40,43,48)(36,45,42,44)(37,47,41,46);(3,10)(4,12)(5,7)(8,16)(17,19,22,30)(18,24,23,27)(20,26,31,28)(21,25,32,29)(33,36,39,42)(34,41,43,37)(35,44,38,45)(40,47,48,46);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,19)(18,24)(20,26)(21,29)(22,30)(23,27)(25,32)(28,31)(33,36)(34,37)(35,45)(38,44)(39,42)(40,47)(41,43)(46,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,30)(27,32)(33,37)(34,42)(35,46)(36,43)(38,47)(39,41)(40,45)(44,48);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,21)(18,26)(19,29)(20,24)(22,32)(23,28)(25,30)(27,31)(33,38,39,35)(34,40,43,48)(36,44,42,45)(37,47,41,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8927267);