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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4127 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)(3,10,13,7)(4,5,15,9)(11,12,14,16)(17,18,23,21)(19,24,25,20)(22,29,30,26)(27,28,32,31)(33,34)(35,45)(36,41)(37,47)(38,42)(39,43)(40,44)(46,48);(1,3,6,13)(2,7,8,10)(4,14,15,11)(5,12,9,16)(17,19)(18,24)(20,21)(22,27)(23,25)(26,31)(28,29)(30,32)(33,35)(34,39)(36,46)(37,38)(40,43)(41,42)(44,45)(47,48);(2,8)(4,15)(7,10)(11,14)(17,20,23,24)(18,25,21,19)(22,28,30,31)(26,32,29,27)(33,36,40,37)(34,41,44,47)(35,46,43,38)(39,42,45,48);(1,4)(2,9)(3,11)(5,8)(6,15)(7,12)(10,16)(13,14)(17,20)(18,19)(21,25)(22,28)(23,24)(26,27)(29,32)(30,31)(33,37,40,36)(34,41,44,47)(35,46,43,38)(39,48,45,42);(1,3)(2,10)(4,14)(5,16)(6,13)(7,8)(9,12)(11,15)(17,21,23,18)(19,24,25,20)(22,26,30,29)(27,28,32,31)(33,38,40,46)(34,42,44,48)(35,37,43,36)(39,41,45,47);(1,5,6,9)(2,4,8,15)(3,12,13,16)(7,14,10,11)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39,40,45)(34,43,44,35)(36,42,37,48)(38,47,46,41)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154549);