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 | |
4099 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18)(19,21)(20,24)(22,23)(25,26)(27,31)(28,29)(30,32)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(2,4)(3,10)(8,12)(11,13)(17,19,22,26)(18,21,23,25)(20,28,30,27)(24,29,32,31)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(2,4)(7,9)(8,12)(15,16)(18,23)(19,26)(24,32)(27,28)(36,45)(40,48)(43,44)(46,47);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,20,22,30)(18,24,23,32)(19,27,26,28)(21,31,25,29)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,19)(18,25)(20,27)(21,23)(22,26)(24,29)(28,30)(31,32)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,21,22,25)(18,19,23,26)(20,29,30,31)(24,28,32,27)(33,37,38,35)(34,39,41,42)(36,44,45,43)(40,46,48,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541546);