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C₂³×M₃₂ #62621

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂³×M₃₂

Subgroups

	subgroup	G/H
center	C₂³×C₈	C₂²
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₈	C₂⁵
socle	C₂⁴	C₂×C₈
prime radical	C₂⁵	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1167 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×A₄×M₃₂, M₃₂×AGL(1,𝔽₈)
maximal subgroup of	#146391, #146392, #146394, …
maximal normal subgroup of	#146391, #146392, #146394, …
maximal quotient of	C₂³×#346, C₂²×C₄×M₃₂, C₂³×C₂²⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55609);