#10143341

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,4)(5,8)(6,7)(9,10,15,17)(11,16,21,23)(12,18,20,14)(13,19,22,24)(25,26,32,36)(27,33,38,40)(28,34,29,31)(30,35,37,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,20)(14,19)(15,21)(17,23)(18,24)(25,27,32,38)(26,33,36,40)(28,39,29,35)(30,31,37,34);(1,4)(2,7)(3,8)(5,6)(10,17)(11,21)(12,20)(19,24)(25,28,32,29)(26,34,36,31)(27,35,38,39)(30,33,37,40);(2,8)(4,6)(9,12,15,20)(10,18,17,14)(11,22,21,13)(16,24,23,19)(25,29,32,28)(26,31,36,34)(27,35,38,39)(30,33,37,40);(1,4)(2,3)(5,6)(7,8)(9,13,15,22)(10,19,17,24)(11,12,21,20)(14,16,18,23)(25,30)(26,35)(27,31)(28,33)(29,40)(32,37)(34,38)(36,39);(1,3)(2,6)(4,8)(5,7)(9,14,15,18)(10,20,17,12)(11,19,21,24)(13,23,22,16)(25,31)(26,28)(27,37)(29,36)(30,38)(32,34)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050537);