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C₂×Q₈×C₁₆ #33472

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×Q₈×C₁₆

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂²
commutator	C₂	C₂³×C₁₆
Frattini, cosocle	C₂×C₈	C₂⁴
socle, prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×C₁₆×SL(2,𝔽₃)
maximal subgroup of	C₁₆×#101, #128508, C₁₆×#103, …
maximal normal subgroup of	C₁₆×#101, #128508, C₁₆×#103, …
maximal quotient of	C₂×C₁₆×C₄⋊C₄, C₄×Q₈×C₁₆, C₂×#13104, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26460);