C₂×#1684 #19122

order	256
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#1684

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#121
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

319 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 256

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1684, #211155, #211175, …
maximal normal subgroup of	C₄×#1684, #211155, #211175, …
maximal quotient of	C₂×#8649, C₂×#9216, C₂×#9220, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12110);