C₂×#2236 #33644

order	256
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#2236

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂⁴	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 315 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)(5,8)(6,7)(9,10)(11,12,13,16)(14,17,20,24)(15,23,21,19)(18,26,25,22);(4,7)(8,10)(12,16)(17,24)(18,25)(19,23);(3,5,6,9)(4,8,7,10)(12,17)(14,22)(15,21)(16,24)(18,19)(20,26)(23,25)

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#137431, #137436, #138973, …
maximal normal subgroup of	#137431, #137436, #138973, …
maximal quotient of	C₂×#12867, C₂×#12919, C₂×#12993, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26632);