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 |
| prime radical | #2456 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3886 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18)(19,24)(20,28)(21,26)(22,30)(23,27)(25,31)(29,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3)(2,7)(4,6)(5,11)(8,15)(9,10)(12,14)(13,16)(17,19,23,22)(18,24,27,30)(20,29,31,26)(21,28,32,25)(34,40)(35,37)(42,46)(43,45);(1,2,6,9)(3,7,4,10)(5,8,14,13)(11,15,12,16)(17,20,23,31)(18,25,27,28)(19,26,22,29)(21,24,32,30)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20)(18,25)(19,26)(21,30)(22,29)(23,31)(24,32)(27,28)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4,6,3)(2,7,9,10)(5,11,14,12)(8,16,13,15)(17,21)(18,26)(19,25)(20,30)(22,28)(23,32)(24,31)(27,29)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,22,23,19)(18,24,27,30)(20,29,31,26)(21,25,32,28)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947929);