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 | |
3773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,21)(19,24,22,30)(20,27,29,25)(26,32,28,31)(33,34)(35,39)(36,42)(37,41)(38,40)(43,45)(44,48)(46,47);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,23,22)(18,24,21,30)(20,26,29,28)(25,31,27,32)(34,35)(37,45)(40,42)(44,46);(2,6)(7,10)(8,12)(14,16)(17,20)(18,25)(19,26)(21,27)(22,28)(23,29)(24,31)(30,32)(33,35,39,34)(36,42,38,40)(37,43,45,41)(44,48,46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(33,36,39,38)(34,40,35,42)(37,44,45,46)(41,47,43,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21,23,18)(19,24,22,30)(20,27,29,25)(26,31,28,32)(33,37,39,45)(34,41,35,43)(36,44,38,46)(40,47,42,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,19)(18,24,21,30)(20,28,29,26)(25,31,27,32)(33,38,39,36)(34,42,35,40)(37,46,45,44)(41,48,43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663294);