C₂×#7142 #126582

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₂×#7142

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₄²
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#128226
maximal normal subgroup of	C₂×#128226
maximal quotient of	C₂×#93249, C₂×#93999

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33778);

C2×#7142 #126582

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7142 #126582