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 | |
4165 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)(3,7)(4,8)(5,11)(6,9)(10,13)(12,15)(14,16)(17,18)(19,22)(20,29)(21,25)(23,27)(24,26)(28,32)(30,31)(33,34,40,37)(35,44,46,41)(36,42,47,38)(39,43,48,45);(1,3,6,11)(2,7,9,5)(4,10,14,12)(8,13,16,15)(17,19)(18,24)(20,28)(21,31)(22,27)(23,26)(25,32)(29,30)(33,35,40,46)(34,41,37,44)(36,45,47,43)(38,39,42,48);(1,3)(2,5)(4,10)(6,11)(7,9)(8,15)(12,14)(13,16)(17,20,23,21)(18,25,27,29)(19,28,26,31)(22,30,24,32)(33,36,40,47)(34,42,37,38)(35,45,46,43)(39,41,48,44);(1,3)(2,5)(4,12)(6,11)(7,9)(8,13)(10,14)(15,16)(17,21,23,20)(18,25,27,29)(19,28,26,31)(22,32,24,30)(33,37,40,34)(35,41,46,44)(36,38,47,42)(39,45,48,43);(1,4,6,14)(2,8,9,16)(3,10,11,12)(5,15,7,13)(17,19)(18,22)(20,28)(21,31)(23,26)(24,27)(25,30)(29,32)(33,38)(34,36)(35,39)(37,47)(40,42)(41,45)(43,44)(46,48);(1,5)(2,3)(4,13)(6,7)(8,12)(9,11)(10,16)(14,15)(17,22)(18,26)(19,27)(20,30)(21,32)(23,24)(25,31)(28,29)(33,39)(34,43)(35,42)(36,41)(37,45)(38,46)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=i,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, 10411806);