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 | |
4259 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(2,7)(3,12)(9,14)(10,13)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,7)(5,16)(8,11)(9,14)(19,26)(23,30)(27,31)(28,32)(33,35,39,36)(34,41,38,40)(37,43,46,44)(42,48,45,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8184036);