C₅×#2843 #10616438

order	640
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#2843

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

348 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 184 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)(6,7,9,11)(8,10,12,13)(14,15)(16,20)(17,21)(18,19);(8,12)(10,13)(15,19)(17,20)(22,23,25,27)(24,28,29,26);(6,7)(8,10)(9,11)(12,13)(14,16)(15,20)(17,19)(18,21)(22,24,25,29)(23,26,27,28);(6,8)(7,10)(9,12)(11,13)(14,17,18,20)(15,21,19,16)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12011889, #12012373, D₅×#2843, …
maximal subgroup of	C₅×#12183, C₅×#12259, C₂×C₅×#2843, …
maximal normal subgroup of	C₅×#12183, C₅×#12259, C₂×C₅×#2843, …
maximal quotient of	C₅×#14638, C₅×#14639, C₅×#14640, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18520);