#136595

order	512
nilpotency class	3
isoclinism	isoclinic class of #9159
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#3035	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fi

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, 43791);