C₂×#1979 #21247

order	256
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#1979

Subgroups

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

719 conjugacy classes of subgroups; only the 247 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 247 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);(4,6)(7,8,12,17,13,16,11,25)(9,14,20,29,19,30,26,28)(10,21,23,31,24,35,22,15)(18,32,34,27,33,37,36,38);(3,4,5,6)(7,9,11,20,13,19,12,26)(8,14,17,28,16,30,25,29)(10,18,22,34,24,33,23,36)(15,27,21,32,31,38,35,37);(7,10,12,23,13,24,11,22)(8,15,25,35,16,31,17,21)(9,18,26,36,19,33,20,34)(14,27,29,37,30,38,28,32)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#147414, #147420, #147435, …
maximal normal subgroup of	#147414, #147420, #147435, …
maximal quotient of	C₂×#9669, C₂×#9671, #204441, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14235);