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 | |
3901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18)(19,29)(20,26)(21,22)(23,31)(24,28)(25,32)(27,30)(33,34,39,42)(35,37,38,40)(36,44,47,41)(43,48,46,45);(3,10)(6,7)(11,16)(14,15)(17,19)(18,25)(20,23)(21,30)(22,31)(24,32)(26,27)(28,29)(33,35,39,38)(34,40,42,37)(36,43,47,46)(41,48,44,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,20,24,21)(18,26,28,22)(19,23,32,30)(25,27,29,31)(33,36)(34,41)(35,43)(37,45)(38,46)(39,47)(40,48)(42,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,21,24,20)(18,26,28,22)(19,30,32,23)(25,27,29,31)(33,37,39,40)(34,35,42,38)(36,45,47,48)(41,43,44,46);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,22,24,26)(18,21,28,20)(19,31,32,27)(23,25,30,29)(33,38,39,35)(34,37,42,40)(36,46,47,43)(41,45,44,48);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,23,24,30)(18,27,28,31)(19,20,32,21)(22,25,26,29)(33,36)(34,41)(35,43)(37,45)(38,46)(39,47)(40,48)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8046701);