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 | |
4333 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)(3,8)(4,11)(5,13)(6,7)(9,15)(10,12)(14,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,3,6,11)(2,4,7,8)(5,10,15,14)(9,12,13,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,7)(4,8)(9,13)(12,16)(18,20)(19,26)(25,28)(27,32)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,4)(2,3)(5,12)(6,8)(7,11)(9,10)(13,14)(15,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,8)(4,7)(5,14)(6,11)(9,12)(10,15)(13,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,13)(8,16)(11,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=i,f^2=gh,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, 10154786);