#129545

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂×#121	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

637 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci,d^b=dgi,f^b=fi,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, 36741);