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 | |
3994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,5)(4,12)(15,16)(17,18)(19,23)(20,29)(21,31)(22,26)(24,30)(25,27)(28,32)(33,34,39,43)(35,37,44,40)(36,41,46,48)(38,45,47,42);(1,2,6,7)(3,8,5,10)(4,13,12,9)(11,16,14,15)(17,18,22,26)(19,23,28,32)(20,24,30,29)(21,25,27,31)(35,44)(36,46)(37,40)(41,48);(1,3,6,5)(2,8,7,10)(4,11,12,14)(9,15,13,16)(17,19,22,28)(18,23,26,32)(20,27,30,21)(24,31,29,25)(33,35)(34,40)(36,45)(37,43)(38,41)(39,44)(42,46)(47,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,36)(34,41)(35,45)(37,47)(38,40)(39,46)(42,44)(43,48);(1,4,6,12)(2,9,7,13)(3,11,5,14)(8,15,10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,37,39,40)(34,35,43,44)(36,47,46,38)(41,45,48,42);(1,5,6,3)(2,10,7,8)(4,14,12,11)(9,16,13,15)(17,21,22,27)(18,25,26,31)(19,20,28,30)(23,24,32,29)(33,38,39,47)(34,42,43,45)(35,41,44,48)(36,40,46,37)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017794);