C₂×#8124 #181405

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#8124

Subgroups

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

1466 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);(3,4,6,10,8,11,7,12)(5,13,15,17,16,9,14,18)(19,20,22,25,23,26,30,33)(21,27,29,34,28,24,31,32);(3,5,7,15,8,16,6,14)(4,9,12,18,11,13,10,17)(19,21)(20,24,26,27)(22,29)(23,28)(25,32,33,34)(30,31);(3,6,8,7)(4,10,11,12)(5,14,16,15)(9,17,13,18)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#181406, C₂×#202152, C₂×#181408
maximal normal subgroup of	C₂×#181406, C₂×#202152, C₂×#181408
maximal quotient of	#72309128

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88601);

C2×#8124 #181405

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#8124 #181405