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C₂×C₄×#97 #20326

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

C₂×C₄×#97

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂²
commutator	C₂	C₂×C₄³
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72280116
maximal subgroup of	C₂×#7019, #95031, #95032, …
maximal normal subgroup of	C₂×#7019, #95031, #95032, …
maximal quotient of	C₂×C₄×#322, C₂×C₈×#97, C₂×#7842, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13314);