C₂×#1306 #8111

order	256
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#1306

Subgroups

	subgroup	G/H
center, socle	C₂³	#95
commutator	C₂³	C₂×C₄²
Frattini, cosocle	C₂³×C₄	C₂³
prime radical	C₂⁵	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 471 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) = 64

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,19,20,16)(14,18)(15,17)(21,25,26,23)(22,24);(3,5)(4,7,8,9)(6,10)(11,13,14,21,15,20,22,26)(12,16,18,23,17,19,24,25)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94292, #94296, C₄×#1306, …
maximal normal subgroup of	#94292, #94296, C₄×#1306, …
maximal quotient of	C₂×#7032, #95111, #95118, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1099);