#225244

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

558 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)(10,14,16,21)(15,17,22,23)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(1,3,4,7)(2,5,6,8)(9,10)(11,17)(12,21)(13,16)(14,18)(15,19)(20,23)(22,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(2,6)(5,8)(27,33)(30,36)(34,39)(37,40);(9,11,18,24,13,20,12,19)(10,15,21,23,16,22,14,17)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132440);