#129515

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	D₄×#41	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

820 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,f^b=fi,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, 36711);