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C₂×C₂²⋊C₃₂ #13552

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂×C₂²⋊C₃₂

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂²
commutator	C₂	C₂²×C₃₂
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂³	C₂×C₁₆
prime radical	C₂⁴	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2);(3,4)(5,6)(7,8,9,13,10,14,17,23,11,15,18,24,20,26,29,33,12,16,19,25,21,27,30,34,22,28,31,35,32,36,37,38);(4,6)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94060, #94063, #94122, …
maximal normal subgroup of	#94060, #94063, #94122, …
maximal quotient of	C₂×#7334, C₂×#7362, #149653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6540);