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C₄×#467 #21522

order	256
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#124179, #124180, #124183, …
maximal normal subgroup of	#124179, #124180, #124183, …
maximal quotient of	C₈×#467, #210724, #210742, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14510);