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 | |
4064 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(33,34,39,44)(35,42,37,40)(36,46,45,41)(38,43,47,48);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,48,46,43);(2,6)(3,10)(8,12)(11,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,36)(34,41)(35,38)(37,47)(39,45)(40,43)(42,48)(44,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,37,39,35)(34,42,44,40)(36,47,45,38)(41,48,46,43);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,38,39,47)(34,43,44,48)(35,45,37,36)(40,46,42,41);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8729106);