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 | #56376 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,8,6,13)(4,9,11,14)(10,15,12,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,28)(29,30)(31,32)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(1,3,7,6)(2,8,5,13)(4,10,11,12)(9,15,14,16)(17,19)(18,24)(20,22)(21,27)(23,28)(25,31)(26,30)(29,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(2,5)(3,6)(4,11)(15,16)(17,20,23,29)(18,25,21,30)(19,22,28,32)(24,31,27,26)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,4)(2,9)(3,10)(5,14)(6,12)(7,11)(8,15)(13,16)(17,21,23,18)(19,27,28,24)(20,25,29,30)(22,31,32,26)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,5,7,2)(3,8,6,13)(4,9,11,14)(10,16,12,15)(17,20)(18,25)(19,22)(21,30)(23,29)(24,31)(26,27)(28,32)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,6,7,3)(2,8,5,13)(4,12,11,10)(9,15,14,16)(17,22)(18,26)(19,20)(21,31)(23,32)(24,30)(25,27)(28,29)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,e^2=i,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, 8922732);