C₂×#2264 #33980

order	256
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#2264

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴
prime radical	C₂×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

231 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

(1,2);(3,4)(5,15)(6,20)(7,26)(8,30)(9,11)(10,16)(12,23)(13,22)(14,28)(17,47)(18,50)(19,42)(21,39)(24,58)(25,41)(27,37)(29,60)(31,43)(32,46)(33,54)(34,53)(35,44)(36,48)(38,65)(40,45)(49,66)(51,56)(52,57)(55,64)(59,62)(61,63);(3,5,9,19)(4,10,11,32)(6,21,25,55)(7,27,29,61)(8,31,28,18)(12,37,35,63)(13,39,36,64)(14,40,30,34)(15,41,42,20)(16,44,46,23)(17,48,49,22)(24,59,57,54)(26,33,60,62)(38,66,51,47)(43,52,50,58)(45,56,53,65);(4,11)(10,32)(12,35)(13,36)(14,30)(15,42)(20,41)(26,60)(33,62)(34,40)(37,63)(38,51)(39,64)(43,50)(47,66)(52,58)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#152532, #152535, #152537, …
maximal normal subgroup of	#152532, #152535, #152537, …
maximal quotient of	C₂×#13618, C₂×#13616, C₂×#13634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26968);