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C₂×#1311 #11397

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂×C₈²
Frattini, cosocle	C₄×C₈	C₂³
socle, prime radical	C₂³	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2);(11,12,13,17,14,18,19,24,15,16,20,22,21,23,25,26);(3,4,5,7,6,8,9,10)(12,16)(17,22)(18,23)(24,26)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72234264, #72234267, #72258137, …
maximal subgroup of	#92896, #92899, #92901, …
maximal normal subgroup of	#92896, #92899, #92901, …
maximal quotient of	C₂×#7052, #122207, #122208, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4385);