C₂×#2141 #33378

order	256
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#2141

Subgroups

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

575 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 231 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)(9,18)(10,15)(11,24)(12,28)(13,17)(14,20)(16,26)(19,32)(21,38)(22,35)(23,33)(25,31)(27,34)(29,36)(30,37);(7,9,13,22)(8,14,17,30)(10,21,12,19)(11,25,27,36)(15,31,28,29)(16,32,33,38)(18,34,35,24)(20,23,37,26);(7,10,13,12)(8,15,17,28)(9,19,22,21)(11,23,27,26)(14,29,30,31)(16,24,33,34)(18,32,35,38)(20,36,37,25)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#146751, #146768, #146773, …
maximal normal subgroup of	#146751, #146768, #146773, …
maximal quotient of	C₂×#11594, C₂×#11662, #481265, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26366);