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 | |
4064 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)(3,11,14,8)(4,9,15,13)(5,12,16,10)(17,18)(19,21)(20,25)(22,32)(23,26)(24,30)(27,29)(28,31)(33,34,39,44)(35,42,37,40)(36,46,45,41)(38,43,47,48);(2,7)(4,15)(8,11)(10,12)(17,19,23,22)(18,21,26,32)(20,28,30,27)(24,29,25,31)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,48,46,43);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,9,16,13)(17,20,23,30)(18,24,26,25)(19,27,22,28)(21,31,32,29)(33,36)(34,41)(35,38)(37,47)(39,45)(40,43)(42,48)(44,46);(1,3)(2,8)(4,10)(5,13)(6,14)(7,11)(9,16)(12,15)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,26)(29,32)(33,37,39,35)(34,42,44,40)(36,47,45,38)(41,48,46,43);(1,4)(2,9)(3,12)(5,8)(6,15)(7,13)(10,14)(11,16)(17,21)(18,19)(20,29)(22,26)(23,32)(24,28)(25,27)(30,31)(33,38,39,47)(34,43,44,48)(35,45,37,36)(40,46,42,41);(1,5)(2,10)(3,13)(4,8)(6,16)(7,12)(9,14)(11,15)(17,22,23,19)(18,21,26,32)(20,27,30,28)(24,29,25,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8924521);