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 | |
4122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,18,22,27)(19,25,26,21)(20,24,31,23)(28,30,29,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19)(18,21)(20,28)(22,26)(23,30)(24,32)(25,27)(29,31)(33,35,39,37)(34,40,42,36)(38,43,46,45)(41,47,48,44);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,20)(18,23)(19,28)(21,30)(22,31)(24,27)(25,32)(26,29)(33,35,39,37)(34,36,42,40)(38,43,46,45)(41,44,48,47);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,20)(18,24)(19,29)(21,30)(22,31)(23,27)(25,32)(26,28)(33,36,39,40)(34,35,42,37)(38,44,46,47)(41,43,48,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,19)(18,25)(20,28)(21,27)(22,26)(23,32)(24,30)(29,31)(33,37,39,35)(34,40,42,36)(38,43,46,45)(41,44,48,47);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,21,22,25)(18,26,27,19)(20,30,31,32)(23,29,24,28)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10362884);