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 | |
3921 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(2,7)(3,10)(4,13)(15,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(2,7)(5,8)(9,12)(15,16)(19,26)(22,24)(27,29)(30,32)(33,35,38,36)(34,40,42,39)(37,44,45,43)(41,46,48,47);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,30,24,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8278446);