C₂×#1381 #12558

order	256
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#1381

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂³	#107
prime radical	C₂⁴	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 38
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94340, #94342, #94344, …
maximal normal subgroup of	#94340, #94342, #94344, …
maximal quotient of	C₂×#7154, #126648, #126685, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5546);