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 | |
4206 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)(3,5)(4,12)(6,9)(7,11)(8,13)(10,16)(14,15)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,31)(29,30)(33,34,39,44)(35,45,47,40)(36,38,43,41)(37,48,46,42);(1,3)(2,7)(4,10)(5,9)(6,11)(8,16)(12,14)(13,15)(17,19,23,27)(18,24,25,22)(20,30,26,28)(21,29,32,31)(33,35)(34,40)(36,37)(38,42)(39,47)(41,48)(43,46)(44,45);(4,13)(8,12)(10,15)(14,16)(18,25)(19,27)(20,26)(29,31)(33,36,39,43)(34,41,44,38)(35,37,47,46)(40,48,45,42);(1,4)(2,8)(3,10)(5,14)(6,13)(7,16)(9,12)(11,15)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,29,24,31)(33,36)(34,41)(35,46)(37,47)(38,44)(39,43)(40,42)(45,48);(1,5,6,7)(2,3,9,11)(4,14,13,16)(8,10,12,15)(17,21)(18,26)(19,29)(20,25)(22,30)(23,32)(24,28)(27,31)(33,37,39,46)(34,42,44,48)(35,36,47,43)(38,45,41,40);(1,3)(2,5)(4,15)(6,11)(7,9)(8,16)(10,13)(12,14)(17,22,23,24)(18,27,25,19)(20,31,26,29)(21,28,32,30)(33,38,39,41)(34,43,44,36)(35,42,47,48)(37,40,46,45)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627955);