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C₄×#330 #8223

order	256
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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

maximal subgroup of	C₈×#330, #96833, #96835, …
maximal normal subgroup of	C₈×#330, #96833, #96835, …
maximal quotient of	C₄×#1271, #96740, #96752, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1211);