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 | |
3954 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)(3,8)(4,12)(5,7)(6,11)(9,14)(10,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(33,34,39,36)(35,42,41,38)(37,40,46,44)(43,48,47,45);(1,3,5,11)(2,6,7,8)(4,10,14,13)(9,15,12,16)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(33,34,39,36)(35,38,41,42)(37,44,46,40)(43,48,47,45);(2,7)(6,8)(9,12)(15,16)(18,20)(19,25)(24,28)(27,29)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,2)(3,6)(4,9)(5,7)(8,11)(10,15)(12,14)(13,16)(17,20,23,18)(19,26,25,22)(21,24,31,28)(27,32,29,30)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,3)(2,8)(4,13)(5,11)(6,7)(9,15)(10,14)(12,16)(17,21)(18,24)(19,27)(20,28)(22,32)(23,31)(25,29)(26,30)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113825);