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 | |
4114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,10,9,5)(4,8,12,15)(11,16,13,14)(33,34)(35,44)(36,41)(37,47)(38,42)(39,43)(40,46)(45,48);(2,7)(3,9)(8,15)(11,13)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35,39,46)(34,40,43,44)(36,45,37,42)(38,47,48,41);(2,7)(4,12)(5,10)(11,13)(18,22)(20,29)(23,25)(26,30)(33,36,39,37)(34,41,43,47)(35,45,46,42)(38,44,48,40);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,16,15,14)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,37,39,36)(34,41,43,47)(35,45,46,42)(38,40,48,44);(1,4,6,12)(2,8,7,15)(3,11,9,13)(5,14,10,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,38,39,48)(34,42,43,45)(35,41,46,47)(36,40,37,44);(1,5)(2,9)(3,7)(4,14)(6,10)(8,13)(11,15)(12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,34)(35,40)(36,47)(37,41)(38,42)(39,43)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947587);