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C₂×#1455 #17332

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

Subgroups

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

497 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#1455⋊C₃
maximal subgroup of	#108601, #126894, C₂×#7225, …
maximal normal subgroup of	#108601, #126894, C₂×#7225, …
maximal quotient of	C₂×#7846, C₄×#1455, C₂×#7852, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10320);