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C₂²×#333 #17696

order	256
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#333

Subgroups

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

667 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#333
maximal subgroup of	#106034, #108336, #110376, …
maximal normal subgroup of	#106034, #108336, #110376, …
maximal quotient of	C₂²×#1272, C₂×#7571, C₂×#7643, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10684);