C₂×#2656 #34571

order	256
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#2656

Subgroups

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

2443 conjugacy classes of subgroups; only the 607 normal subgroups are shown.

normal subgroups only

This group has 2443 conjugacy classes of subgroups — too many to draw. Only its 607 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) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(3,4,5,6)(11,12)(13,15)(14,16)(17,18);(7,8,9,10)(11,12,14,16)(13,15,18,17);(3,4)(5,6)(7,8)(9,10)(11,12)(13,17)(14,16)(15,18);(11,13)(12,15)(14,18)(16,17)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#2656⋊C₃, #72228207, #72228213, …
maximal subgroup of	#111266, #112268, #133119, …
maximal normal subgroup of	#111266, #112268, #133119, …
maximal quotient of	C₂×#14271, C₂×#14321, C₂×#14322, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27559);