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C₂×#1273 #7847

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂²
commutator	C₂	C₂×C₈²
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂⁴	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

491 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2);(3,4)(5,6)(7,8,9,11,10,12,13,14);(4,6)(15,16,17,19,18,20,21,22)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#1273⋊C₃
maximal subgroup of	#93274, #93275, #93323, …
maximal normal subgroup of	#93274, #93275, #93323, …
maximal quotient of	C₂×#7014, #94863, C₂×#7015, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 835);