#8936582

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

4373 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:

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

Polycyclic presentation:

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

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, 8843778);