C₂×#2986 #60246

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#2986

Subgroups

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

897 conjugacy classes of subgroups; only the 719 normal subgroups are shown.

normal subgroups only

This group has 897 conjugacy classes of subgroups — too many to draw. Only its 719 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) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,22,25,23,26,29,32)(21,24,28,31,27,30,33,34);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,24)(22,28)(23,27)(25,31)(26,30)(29,33)(32,34);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(21,27)(24,30)(28,33)(31,34);(4,10)(8,11)(9,18)(16,17)(21,27)(24,30)(28,33)(31,34)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#106048, #106051, #106177, …
maximal normal subgroup of	#106048, #106051, #106177, …
maximal quotient of	C₂×#20538, C₂×#20540, C₂×#20571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53234);