C₂×#1366 #11955

order	256
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#1366

Subgroups

	subgroup	G/H
center, socle	C₂²	#333
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂³	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

147 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 147 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1366, #128924, #128945, …
maximal normal subgroup of	C₄×#1366, #128924, #128945, …
maximal quotient of	C₂×#7135, #124241, #124983, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4943);