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 | |
4178 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)(3,5,11,7)(4,8,12,15)(10,13,16,14)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(4,12)(8,15)(10,16)(13,14)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(34,40)(35,37)(42,46)(43,45);(1,3,6,11)(2,7,9,5)(4,10,12,16)(8,14,15,13)(17,18,21,24)(19,25,27,22)(20,23,30,28)(26,32,29,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,19)(18,22)(20,26)(21,27)(23,31)(24,25)(28,32)(29,30)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5)(2,3)(4,13)(6,7)(8,10)(9,11)(12,14)(15,16)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150485);