C₂×#3483 #62716

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#3483

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
socle	C₂³	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

4209 conjugacy classes of subgroups; only the 2943 normal subgroups are shown.

normal subgroups only

This group has 4209 conjugacy classes of subgroups — too many to draw. Only its 2943 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#85604, C₂×#85654, #72261377
maximal subgroup of	#185461, #185489, #185729, …
maximal normal subgroup of	#185461, #185489, #185729, …
maximal quotient of	C₂×#34129, C₂×#34155, C₂×C₄×#550, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55704);