C₄×#1938 #354362

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₄×#1938

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#21546, D₄×#1938, #72310151
maximal normal subgroup of	C₄×#21546, D₄×#1938, #72310151

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261558);

C4×#1938 #354362

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×#1938 #354362