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 | |
4249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,11,8)(5,13,15,9)(10,16,14,12)(17,18)(19,26)(20,30)(21,24)(22,31)(23,25)(27,32)(28,29)(33,34,39,44)(35,42,37,40)(36,41,43,38)(45,46,47,48);(2,7)(4,8)(9,13)(12,16)(17,19,23,21)(18,24,25,26)(20,28,31,27)(22,29,30,32)(35,37)(40,42)(45,47)(46,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,16)(12,13)(14,15)(17,20)(18,22)(19,27)(21,28)(23,31)(24,32)(25,30)(26,29)(33,35,39,37)(34,40,44,42)(36,45,43,47)(38,46,41,48);(1,3)(2,8)(4,7)(5,14)(6,11)(9,12)(10,15)(13,16)(17,20)(18,22)(19,28)(21,27)(23,31)(24,29)(25,30)(26,32)(33,36)(34,41)(35,45)(37,47)(38,44)(39,43)(40,48)(42,46);(1,4,6,8)(2,3,7,11)(5,12,15,16)(9,10,13,14)(17,21,23,19)(18,24,25,26)(20,28,31,27)(22,32,30,29)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,5)(2,9)(3,10)(4,12)(6,15)(7,13)(8,16)(11,14)(17,22)(18,20)(19,29)(21,32)(23,30)(24,28)(25,31)(26,27)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329564);