#136631

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂×#466	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

514 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321045, #72323077, #72323090
maximal normal subgroup of	#72321045, #72323077, #72323090
maximal quotient of	#72321075

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43827);