C₄×#2104 #359419

order	512
nilpotency class	3
isoclinism	isoclinic class of #2104
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₄×#2104

Subgroups

	subgroup	G/H
center	C₂×C₄	#401
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#401
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 943 conjugacy classes of subgroups — too many to draw. Only its 252 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266615);