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C₄×#322 #7839

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄³

Aliases

Subgroups

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

327 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#93258, #93266, #93315, …
maximal normal subgroup of	#93258, #93266, #93315, …
maximal quotient of	C₈×#322, #94853, #94855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 827);