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 | |
4178 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18,24,22)(19,21,25,20)(23,30,31,26)(27,28,32,29)(33,34,39,37)(35,40,46,45)(36,42,47,41)(38,48,44,43);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19)(18,20)(21,22)(23,27)(24,25)(26,28)(29,30)(31,32)(33,35,39,46)(34,40,37,45)(36,44,47,38)(41,48,42,43);(1,2,6,4)(3,8,11,7)(5,9,13,12)(10,15,16,14)(17,20,24,21)(18,25,22,19)(23,28,31,29)(26,32,30,27)(33,36)(34,41)(35,44)(37,42)(38,46)(39,47)(40,48)(43,45);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,21,24,20)(18,25,22,19)(23,29,31,28)(26,32,30,27)(33,36)(34,42)(35,44)(37,41)(38,46)(39,47)(40,43)(45,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,22,24,18)(19,21,25,20)(23,26,31,30)(27,28,32,29)(33,37,39,34)(35,45,46,40)(36,42,47,41)(38,48,44,43);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,23)(18,26)(19,27)(20,28)(21,29)(22,30)(24,31)(25,32)(33,38,39,44)(34,43,37,48)(35,36,46,47)(40,41,45,42)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,e^2=hi,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, 10413474);