C₄×#1778 #353227

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

direct product

C₄×#1778

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#20512, C₄×#13138
maximal normal subgroup of	C₄×#20512, C₄×#13138

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260423);

C4×#1778 #353227

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×#1778 #353227