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 | |
4207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,16)(4,9,13,12)(5,14,15,10)(18,23)(19,25)(24,30)(26,31)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(1,3,6,11)(2,7,8,16)(4,10,13,14)(5,12,15,9)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(34,36)(39,41)(40,44)(47,48);(2,8)(3,11)(4,13)(5,15)(19,25)(20,27)(21,29)(24,30)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,12)(11,14)(15,16)(17,19,22,25)(18,20,23,27)(21,26,29,31)(24,28,30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,5,6,15)(2,10,8,14)(3,12,11,9)(4,7,13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=ghi,e^2=hi,f^2=h,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, 8924788);