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C₂²×#335 #17784

order	256
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#335

Subgroups

	subgroup	G/H
center	C₂³×C₄	D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₄²	C₂⁴
socle, prime radical	C₂⁴	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

471 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);(6,9)(7,10)(11,12)(13,14,15,17,16,18,19,20);(5,6,10,12,8,11,7,9)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#335
maximal subgroup of	#108842, #108843, #108845, …
maximal normal subgroup of	#108842, #108843, #108845, …
maximal quotient of	C₂²×#1277, C₂×C₄×#335, C₂×#8596, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10772);