C₂×#1636 #18491

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×#1636

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#120
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

457 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 457 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);(3,4,8,11)(5,12,14,17)(6,10,7,15)(9,16,18,13)(19,20,25,29)(21,30,22,33)(23,28,24,27)(26,32,34,31);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(19,21,25,22)(20,26,29,34)(23,32,24,31)(27,33,28,30);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,22,25,21)(20,26,29,34)(23,31,24,32)(27,33,28,30)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1636, #209721, #209769, …
maximal normal subgroup of	C₄×#1636, #209721, #209769, …
maximal quotient of	C₂×#7763, C₂×#7827, C₂×#7771, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11479);