C₂×#2047 #23519

order	256
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#2047

Subgroups

	subgroup	G/H
center, socle	C₂³	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂⁴	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

665 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 665 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(3,4)(5,6)(7,8,12,16)(9,17,20,32)(10,21,24,14)(11,25,27,15)(13,26,29,19)(18,34,22,33)(23,31,38,36)(28,35,30,37);(7,9,12,20)(8,13,16,29)(10,22,24,18)(11,26,27,19)(14,30,21,28)(15,32,25,17)(23,35,38,37)(31,33,36,34);(4,6)(7,10,11,23,12,24,27,38)(8,14,15,31,16,21,25,36)(9,18,19,35,20,22,26,37)(13,28,17,33,29,30,32,34)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#137188, #137194, #142281, …
maximal normal subgroup of	#137188, #137194, #142281, …
maximal quotient of	C₂×#10516, C₂×#10517, #238811, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16507);