C₂×#2843 #36690

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×#2843

Subgroups

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

903 conjugacy classes of subgroups; only the 487 normal subgroups are shown.

normal subgroups only

This group has 903 conjugacy classes of subgroups — too many to draw. Only its 487 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) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#2843⋊C₃, #72278643
maximal subgroup of	#131483, C₂×#12183, C₂×#12259, …
maximal normal subgroup of	#131483, C₂×#12183, C₂×#12259, …
maximal quotient of	C₂×#14638, C₂×#14639, C₂×#14640, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29678);