C₂×#1703 #19141

order	256
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#1703

Subgroups

	subgroup	G/H
center, socle	C₂³	#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂²×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

347 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,15,18)(13,19,21,26)(14,23,20,17)(16,24,25,22);(4,7)(8,9)(11,13,20,24,15,21,14,22)(12,16,23,26,18,25,17,19);(4,8)(6,10)(7,9)(13,21)(16,25)(19,26)(22,24)

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1703, #211441, #211461, …
maximal normal subgroup of	C₄×#1703, #211441, #211461, …
maximal quotient of	C₂×#8656, C₂×#9223, C₂×#9227, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12129);