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C₂×#1842 #20789

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 1572864
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Subgroups

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

735 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);(4,6)(8,10)(11,12,13,15,14,16,17,18);(3,4,5,6);(7,8,9,10)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#1842⋊C₃
maximal subgroup of	C₂×#7040, #95227, #95240, …
maximal normal subgroup of	C₂×#7040, #95227, #95240, …
maximal quotient of	C₂×#8805, C₂×#8806, C₂×#8807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13777);