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 | |
4081 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)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,29,32,31)(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)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,31,28,29)(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)(17,18,22,26)(19,23,25,21)(20,28,30,24)(27,29,32,31)(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)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32)(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)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(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)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,32,28,27)(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=gh,c^2=gi,d^2=hi,e^2=i,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, 8922567);