C₂×#1638 #18493

order	256
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#1638

Subgroups

	subgroup	G/H
center, socle	C₂³	#120
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

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

normal subgroups only

This group has 473 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) = 128

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

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1638, #209729, #209777, …
maximal normal subgroup of	C₄×#1638, #209729, #209777, …
maximal quotient of	C₂×#7765, C₂×#7828, C₂×#7775, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11481);