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 | |
3941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,18,23,27)(19,22,29,24)(20,30,28,25)(21,31,32,26)(33,34)(35,44)(36,37)(38,46)(39,40)(41,45)(42,48)(43,47);(1,2,6,3)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,19,23,29)(18,24,27,22)(20,21,28,32)(25,31,30,26)(34,40)(35,45)(37,42)(43,46);(2,3)(7,9)(8,10)(14,15)(17,20,23,28)(18,25,27,30)(19,21,29,32)(22,26,24,31)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,21,23,32)(18,26,27,31)(19,28,29,20)(22,30,24,25)(33,36)(34,42)(35,46)(37,40)(38,41)(39,48)(43,45)(44,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,24)(20,21)(22,27)(23,29)(25,31)(26,30)(28,32)(33,37,39,42)(34,36,40,48)(35,47,45,38)(41,46,44,43);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,24)(18,19,27,29)(20,26,28,31)(21,30,32,25)(33,38,39,47)(34,43,40,46)(35,42,45,37)(36,41,48,44)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=hi,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8780960);