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 | |
4057 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)(3,8)(4,10)(5,11)(6,7)(9,13)(12,16)(14,15)(17,18)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(1,3,7,15)(2,8,6,14)(4,12,13,5)(9,11,10,16)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(1,4,7,13)(2,9,6,10)(3,5,15,12)(8,16,14,11)(17,20,23,30)(18,22,25,29)(19,26,28,32)(21,31,24,27)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,4)(2,10)(3,12)(5,15)(6,9)(7,13)(8,16)(11,14)(17,20)(18,22)(19,26)(21,31)(23,30)(24,27)(25,29)(28,32)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,5,7,12)(2,11,6,16)(3,13,15,4)(8,9,14,10)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,6,7,2)(3,14,15,8)(4,9,13,10)(5,16,12,11)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gi,d^2=ghi,e^2=gi,f^2=gh,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, 8922589);