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 | |
4169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,12)(3,10,14,7)(4,9,15,8)(5,13,16,11)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,3)(2,7)(4,11)(5,8)(6,14)(9,16)(10,12)(13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,35,39,37)(34,40,42,36)(38,43,46,45)(41,47,48,44);(1,4)(2,8)(3,11)(5,7)(6,15)(9,12)(10,16)(13,14)(17,20)(18,24)(19,21)(22,29)(23,25)(26,31)(27,32)(28,30)(33,35,39,37)(34,36,42,40)(38,43,46,45)(41,44,48,47);(1,4)(2,9)(3,13)(5,7)(6,15)(8,12)(10,16)(11,14)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,36,39,40)(34,35,42,37)(38,44,46,47)(41,43,48,45);(1,3)(2,10)(4,11)(5,9)(6,14)(7,12)(8,16)(13,15)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,37,39,35)(34,40,42,36)(38,43,46,45)(41,44,48,47);(1,5)(2,11)(3,8)(4,7)(6,16)(9,14)(10,15)(12,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,38,39,46)(34,41,42,48)(35,43,37,45)(36,44,40,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,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, 10411692);