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 | |
4143 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)(3,8)(4,6)(5,10)(7,9)(11,13)(12,15)(14,16)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34,39,44)(35,43,38,40)(36,46,47,41)(37,42,48,45);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,15,16,13)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(33,35,39,38)(34,40,44,43)(36,45,47,42)(37,46,48,41);(2,9)(3,4)(10,16)(11,12)(18,25)(19,28)(20,30)(26,32)(33,36,39,47)(34,41,44,46)(35,45,38,42)(37,43,48,40);(1,4,7,3)(2,6,9,8)(5,11,14,12)(10,13,16,15)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,36)(34,41)(35,45)(37,40)(38,42)(39,47)(43,48)(44,46);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,15,8,13)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,37)(34,42)(35,41)(36,43)(38,46)(39,48)(40,47)(44,45);(1,6)(2,3)(4,9)(5,13)(7,8)(10,12)(11,16)(14,15)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,38,39,35)(34,43,44,40)(36,42,47,45)(37,41,48,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10103144);