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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3837 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)(17,18,23,26)(19,24,21,22)(20,25,28,32)(27,31,30,29)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,21)(18,24,26,22)(20,27,28,30)(25,31,32,29)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(2,6)(3,10)(4,13)(14,15)(19,21)(20,28)(22,24)(25,32)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,23,19)(18,22,26,24)(20,27,28,30)(25,31,32,29)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22)(18,19)(20,29)(21,26)(23,24)(25,27)(28,31)(30,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8914859);