#8689143

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

4561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8596339);