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 | |
3910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,13)(4,9,14,16)(5,12,15,10)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,40,44)(35,41,46,48)(36,42,37,39)(38,43,45,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,40,46)(34,41,44,48)(36,47,37,43)(38,39,45,42);(2,8)(3,11)(9,16)(10,12)(18,20)(19,26)(25,28)(27,32)(33,36,40,37)(34,42,44,39)(35,43,46,47)(38,41,45,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,37,40,36)(34,39,44,42)(35,43,46,47)(38,41,45,48);(1,4,6,14)(2,9,8,16)(3,12,11,10)(5,7,15,13)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,38)(34,43)(35,39)(36,41)(37,48)(40,45)(42,46)(44,47);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,13,14,7)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,39)(34,36)(35,45)(37,44)(38,46)(40,42)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050927);