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 | |
4339 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)(3,7,12,5)(4,8,14,16)(10,15,11,13)(17,18)(19,24)(20,29)(21,25)(22,28)(23,26)(27,31)(30,32)(33,34,38,41)(35,39,36,43)(37,40,46,45)(42,47,44,48);(1,3,6,12)(2,7,9,5)(4,11,14,10)(8,13,16,15)(17,19)(18,24)(20,30)(21,27)(22,26)(23,28)(25,31)(29,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,20,23,21)(18,25,26,29)(19,27,28,30)(22,31,24,32)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(17,21,23,20)(18,25,26,29)(19,27,28,30)(22,32,24,31)(33,34)(35,39)(36,43)(37,45)(38,41)(40,46)(42,48)(44,47);(1,3)(2,5)(4,10)(6,12)(7,9)(8,13)(11,14)(15,16)(17,22)(18,19)(20,31)(21,32)(23,24)(25,27)(26,28)(29,30)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,5)(2,3)(4,13)(6,7)(8,10)(9,12)(11,16)(14,15)(17,19)(18,22)(20,27)(21,30)(23,28)(24,26)(25,31)(29,32)(33,37,38,46)(34,40,41,45)(35,42,36,44)(39,47,43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=g,f^2=g,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, 10327386);