C₅×#2809 #10616404

order	640
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#2809

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

368 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 640

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11988050, #11988143, #11988219, …
maximal subgroup of	C₅×#7743, C₅×#7760, C₅×#7771, …
maximal normal subgroup of	C₅×#7743, C₅×#7760, C₅×#7771, …
maximal quotient of	C₅×#14446, C₅×#14464, C₅×#14467, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18486);