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 | |
4128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,7,11)(5,8,14,16)(10,12,13,15)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,32,28,25)(33,34,39,36)(35,42,40,37)(38,44,46,41)(43,45,47,48);(1,3)(2,4)(5,13)(6,7)(8,15)(9,11)(10,14)(12,16)(17,19,22,30)(18,23,26,27)(20,28,29,21)(24,25,31,32)(35,40)(37,42)(38,46)(41,44);(3,7)(4,11)(10,13)(12,15)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,25,27,32)(33,35,39,40)(34,37,36,42)(38,43,46,47)(41,45,44,48);(1,4,6,11)(2,7,9,3)(5,12,14,15)(8,13,16,10)(17,20)(18,24)(19,28)(21,30)(22,29)(23,25)(26,31)(27,32)(33,36,39,34)(35,37,40,42)(38,44,46,41)(43,45,47,48);(1,2)(3,4)(5,8)(6,9)(7,11)(10,12)(13,15)(14,16)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,24,27,31)(33,37)(34,40)(35,36)(38,45)(39,42)(41,47)(43,44)(46,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,13)(9,16)(11,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,32)(29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746592);