C₅×#1528 #10603036

order	640
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#1528

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₅
prime radical	C₂³×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

212 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11990940, #11990945, #11990950, …
maximal subgroup of	C₅×#8371, C₅×#8372, C₅×#8373, …
maximal normal subgroup of	C₅×#8371, C₅×#8372, C₅×#8373, …
maximal quotient of	C₅×#7573, C₅×#7576, C₅×#7585, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5118);