#351472

order	512
nilpotency class	3
isoclinism	isoclinic class of #348438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,g^d=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258668);