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C₂²×#330 #17634

order	256
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×#330

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

579 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);(6,9,10,11)(13,14,17,19,18,15,16,20);(5,6)(7,11)(8,10)(9,12)(13,15,18,14)(16,19,17,20)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#330
maximal subgroup of	#106030, #107358, #110372, …
maximal normal subgroup of	#106030, #107358, #110372, …
maximal quotient of	C₂²×#1271, C₂×#7568, C₂×#7627, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10622);