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C₄×#484 #21575

order	256
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂²×D₄
prime radical	C₂⁴	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#99586, #99594, C₄×#1614, …
maximal normal subgroup of	#99586, #99594, C₄×#1614, …
maximal quotient of	C₈×#484, #212027, #212055, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14563);