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 | |
3958 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18,23,26)(19,24,29,20)(21,22,25,31)(27,28,32,30)(35,43)(36,45)(39,46)(40,48);(2,7)(3,12)(9,13)(10,16)(17,19,23,29)(18,24,26,20)(21,27,25,32)(22,28,31,30)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(2,7)(4,14)(8,11)(10,16)(17,20,23,24)(18,19,26,29)(21,30,25,28)(22,27,31,32)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,15)(14,16)(17,21)(18,22)(19,27)(20,30)(23,25)(24,28)(26,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,11,15,8)(17,20)(18,19)(21,30)(22,27)(23,24)(25,28)(26,29)(31,32)(33,36,38,45)(34,40,42,48)(35,44,43,37)(39,47,46,41);(1,5)(2,10)(3,13)(4,11)(6,15)(7,16)(8,14)(9,12)(17,22)(18,25)(19,28)(20,27)(21,26)(23,31)(24,32)(29,30)(33,37,38,44)(34,41,42,47)(35,45,43,36)(39,48,46,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440750);