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C₄×#365 #12669

order	256
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	#106
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#106
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#365, #129160, #129171, …
maximal normal subgroup of	C₈×#365, #129160, #129171, …
maximal quotient of	C₄×#1368, #129105, #129127, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5657);