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 | |
4179 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,29)(28,32)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,46,48,44);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,27)(18,23,26,29)(20,21,28,31)(24,30,32,25)(33,35,38,40)(34,39,42,36)(37,43,47,48)(41,46,45,44);(2,3)(7,9)(8,10)(14,15)(19,27)(20,28)(23,29)(24,32)(33,36,38,39)(34,40,42,35)(37,44,47,46)(41,48,45,43);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,24)(19,21)(22,28)(23,30)(25,29)(26,32)(27,31)(33,36)(34,35)(37,46)(38,39)(40,42)(41,48)(43,45)(44,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,37,38,47)(34,41,42,45)(35,43,40,48)(36,44,39,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,18)(19,29)(20,24)(21,25)(22,26)(23,27)(28,32)(30,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9287061);