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 | |
4248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,4,10)(5,13,15,11)(6,14,16,12)(21,29)(24,30)(26,31)(27,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3)(2,8)(4,7)(5,12)(6,13)(9,10)(11,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(34,35)(39,41)(40,42)(46,47);(2,9)(3,4)(11,13)(12,14)(18,23)(19,25)(24,30)(26,31)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066774);