C₂×#7884 #180948

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#7884

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240032, #72240048, C₂×#109996
maximal normal subgroup of	#72240032, #72240048, C₂×#109996
maximal quotient of	#72240037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88144);

C2×#7884 #180948

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7884 #180948