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 | |
3943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,12)(4,13,14,9)(5,10,16,15)(17,18)(19,28)(20,25)(21,32)(22,27)(23,30)(24,31)(26,29)(35,42)(37,40)(43,48)(45,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,12)(13,16)(14,15)(17,19)(18,23)(20,21)(22,30)(24,32)(25,26)(27,28)(29,31)(33,34,39,38)(35,40,42,37)(36,44,46,41)(43,45,48,47);(2,8)(3,11)(9,13)(10,15)(17,20,22,31)(18,24,27,25)(19,21,30,29)(23,32,28,26)(33,35,39,42)(34,40,38,37)(36,43,46,48)(41,47,44,45);(1,2)(3,12)(4,9)(5,15)(6,8)(7,11)(10,16)(13,14)(17,20)(18,25)(19,29)(21,30)(22,31)(23,32)(24,27)(26,28)(33,36,39,46)(34,41,38,44)(35,43,42,48)(37,45,40,47);(1,4)(2,9)(3,10)(5,7)(6,14)(8,13)(11,15)(12,16)(17,21,22,29)(18,26,27,32)(19,20,30,31)(23,25,28,24)(33,37)(34,35)(36,45)(38,42)(39,40)(41,43)(44,48)(46,47);(1,5,6,16)(2,10,8,15)(3,9,11,13)(4,12,14,7)(17,18)(19,23)(20,24)(21,32)(22,27)(25,31)(26,29)(28,30)(33,38,39,34)(35,37,42,40)(36,44,46,41)(43,45,48,47)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=hi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653143);