C₂×#2834 #36656

order	256
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#2834

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂⁵	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

979 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 979 conjugacy classes of subgroups — too many to draw. Only its 447 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 — 5 generators on 26 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#2834⋊C₃, #72261303
maximal subgroup of	#131556, C₂×#12138, C₂×#12232, …
maximal normal subgroup of	#131556, C₂×#12138, C₂×#12232, …
maximal quotient of	C₂×#14627, C₂×#14636, C₂×#14644, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29644);