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:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(18,21)(19,20)(26,29)(27,28)(33,34)(35,37)(36,46)(38,40)(39,45)(41,43)(42,48)(44,47);(3,10)(6,7)(11,16)(14,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,35)(34,41)(36,45)(37,38)(39,42)(40,43)(44,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,36)(34,42)(35,45)(37,47)(38,46)(39,41)(40,48)(43,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,37,40,41)(34,43,38,35)(36,47,48,39)(42,44,46,45);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,38,40,34)(35,37,43,41)(36,46,48,42)(39,45,47,44);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,39,40,47)(34,44,38,45)(35,42,43,46)(36,41,48,37)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8046672);