C₅×#2640 #10616235

order	640
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₅×#2640

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

500 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12000656, #12000670, #12000712, …
maximal subgroup of	C₅×#10090, C₅×#10092, C₅×#10098, …
maximal normal subgroup of	C₅×#10090, C₅×#10092, C₅×#10098, …
maximal quotient of	C₅×#14260, C₅×#14261, C₅×#14273, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18317);