C₂×#1678 #19119

order	256
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#1678

Subgroups

	subgroup	G/H
center, socle	C₂³	#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

361 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(4,7)(5,9)(8,10)(11,12)(13,19)(14,16)(15,23)(17,21)(18,22)(20,26)(24,25);(3,4,5,7,6,8,9,10)(12,16)(17,25)(18,23)(19,26);(11,13,14,20)(12,17,16,25)(15,24,22,21)(18,26,23,19)

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1678, #211150, #211170, …
maximal normal subgroup of	C₄×#1678, #211150, #211170, …
maximal quotient of	C₂×#8647, C₂×#9216, C₂×#9222, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12107);