C₂×#1676 #19117

order	256
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#1676

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 143 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 143 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)(19,20,23,26)(21,27,29,34)(22,30,32,25)(24,31,33,28);(3,5,12,16,7,14,6,13)(4,8,15,18,10,17,9,11);(19,21,23,29)(20,24,26,33)(22,31,32,28)(25,34,30,27)

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#136997, #136998, #137004, …
maximal normal subgroup of	#136997, #136998, #137004, …
maximal quotient of	C₂×#8645, C₂×#9216, C₂×#9568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12105);