#141857

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=defg,c^2=ei,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gi,h^b=hi

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, 49053);