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C₂×#1357 #11918

order	256
nilpotency class	4
isoclinism	isoclinic class of #1357
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	#331
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂×#493	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

247 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94779, C₄×#1357, #128916, …
maximal normal subgroup of	#94779, C₄×#1357, #128916, …
maximal quotient of	C₂×#7092, #122857, #122859, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4906);