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 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18,22,25)(19,21,27,23)(20,24,28,31)(26,29,32,30)(33,34)(35,45)(36,41)(37,47)(38,42)(39,43)(40,44)(46,48);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(20,28)(24,31)(26,32)(29,30)(33,35)(34,39)(36,46)(37,38)(40,43)(41,42)(44,45)(47,48);(2,7)(3,10)(8,13)(11,16)(17,19,22,27)(18,23,25,21)(20,26,28,32)(24,30,31,29)(33,36,40,37)(34,41,44,47)(35,46,43,38)(39,42,45,48);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31)(33,37,40,36)(34,41,44,47)(35,46,43,38)(39,48,45,42);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,29,23,30)(33,38,40,46)(34,42,44,48)(35,37,43,36)(39,41,45,47);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,30)(31,32)(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=i,c^2=gh,d^2=gi,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, 10154186);