C₂×#1637 #18489

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×#1637

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

587 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 151 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);(3,4,6,9)(5,7,10,8)(11,12,17,21)(13,22,14,25)(15,20,16,19)(18,24,26,23);(4,7)(5,10)(8,9)(11,13,17,14)(12,18,21,26)(15,24,16,23)(19,25,20,22);(11,14,17,13)(12,18,21,26)(15,23,16,24)(19,25,20,22)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1637, #209725, #209773, …
maximal normal subgroup of	C₄×#1637, #209725, #209773, …
maximal quotient of	C₂×#7764, C₂×#7774, C₂×#7823, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11477);