#140844

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×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	C₂³	C₂×#102
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 48040);