C₂×#2239 #33655

order	256
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#2239

Subgroups

	subgroup	G/H
center, socle	C₂²	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂⁴	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#137447, #137450, #137452, …
maximal normal subgroup of	#137447, #137450, #137452, …
maximal quotient of	C₂×#12875, C₂×#12931, C₂×#13009, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26643);