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 | #2669 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,14)(4,13,15,9)(5,10,16,12)(33,34,39,44)(35,45,47,40)(36,41,46,42)(37,43,48,38);(1,3,6,11)(2,7,8,14)(4,12,15,10)(5,9,16,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35)(34,40)(36,43)(37,41)(38,46)(39,47)(42,48)(44,45);(2,8)(3,11)(9,13)(10,12)(18,23)(19,27)(25,29)(26,28)(33,36,39,46)(34,41,44,42)(35,43,47,38)(37,45,48,40);(1,3)(2,7)(4,12)(5,13)(6,11)(8,14)(9,16)(10,15)(17,19,22,27)(18,24,23,31)(20,28,30,26)(21,25,32,29)(33,36)(34,42)(35,43)(37,45)(38,47)(39,46)(40,48)(41,44);(1,4,6,15)(2,9,8,13)(3,12,11,10)(5,7,16,14)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,32)(26,27)(33,37,39,48)(34,43,44,38)(35,41,47,42)(36,45,46,40);(1,5,6,16)(2,10,8,12)(3,9,11,13)(4,14,15,7)(17,21,22,32)(18,26,23,28)(19,29,27,25)(20,24,30,31)(33,38,39,43)(34,37,44,48)(35,46,47,36)(40,41,45,42)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8632045);