C₂×#2216 #33598

order	256
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#2216

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

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

normal subgroups only

This group has 351 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) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(3,4)(5,6)(7,8,12,16)(9,17,21,33)(10,22,24,37)(11,25,23,15)(13,26,28,19)(14,29,30,18)(20,36,35,32)(27,34,31,38);(7,9,18,34,11,20,24,26,12,21,29,38,23,35,10,19)(8,13,22,32,15,27,30,33,16,28,37,36,25,31,14,17);(4,6)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#141452, #141456, #141458, …
maximal normal subgroup of	#141452, #141456, #141458, …
maximal quotient of	C₂×#12431, C₂×#12452, C₂×#12469, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26586);