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C₂×C₄×#131 #60055

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

C₂×C₄×#131

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂³×C₄²
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂³	C₂³×C₄
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×C₄×#131⋊C₃
maximal subgroup of	#105634, #105672, #105968, …
maximal normal subgroup of	#105634, #105672, #105968, …
maximal quotient of	C₂×C₈×#131, C₂×#17320, C₂×#17328, …
automorphism group of	C₂×C₅×C₃₂, C₅×M₆₄
outer automorphism group of	C₂×C₅×C₃₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53043);