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 | |
4087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,4)(6,7)(9,10,13,16)(11,14,17,18)(12,15,21,20)(19,22,23,24)(26,31)(29,41)(32,35)(33,42)(37,51)(40,52)(44,47)(50,56);(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(28,39)(33,42)(36,49)(40,52)(44,47)(46,53)(50,56)(54,55);(1,3,5,8)(2,6,4,7)(11,17)(14,18)(19,23)(22,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,47)(37,48)(39,42)(40,46)(41,43)(44,49)(45,51)(50,55)(52,53)(54,56);(1,4,5,2)(3,7,8,6)(9,10)(11,18)(12,20)(13,16)(14,17)(15,21)(19,22)(23,24)(25,27,30,38)(26,32,31,35)(28,36,39,49)(29,37,41,51)(33,44,42,47)(34,45,43,48)(40,50,52,56)(46,54,53,55);(9,11,13,17)(10,14,16,18)(12,19,21,23)(15,22,20,24)(25,28,30,39)(26,33,31,42)(27,36,38,49)(29,40,41,52)(32,44,35,47)(34,46,43,53)(37,50,51,56)(45,54,48,55);(9,12)(10,15)(11,19)(13,21)(14,22)(16,20)(17,23)(18,24)(25,29,30,41)(26,34,31,43)(27,37,38,51)(28,40,39,52)(32,45,35,48)(33,46,42,53)(36,50,49,56)(44,54,47,55)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575458);