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C₂×#1309 #8127

order	256
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	#95
commutator	C₂³	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂⁴	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2);(3,4,5,8,6,7,9,10)(12,15)(13,17);(4,7)(5,9)(11,12,13,16,14,15,17,18)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#1309⋊C₃
maximal subgroup of	#94331, #94337, C₄×#1309, …
maximal normal subgroup of	#94331, #94337, C₄×#1309, …
maximal quotient of	C₂×#7034, #95150, #95164, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1115);