C₅×#2571 #10616166

order	640
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#2571

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

380 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 216 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)(6,7,10,8)(9,12,13,11)(15,17)(19,20)(22,23,26,29)(24,28,25,27);(7,8)(11,12)(14,15,18,17)(16,19,21,20)(22,24,26,25)(23,27,29,28);(6,8,10,7)(9,12,13,11)(14,16,18,21)(15,19,17,20)(22,23)(24,27)(25,28)(26,29);(6,9,10,13)(7,11,8,12)(14,17,18,15)(16,20,21,19)(22,25,26,24)(23,28,29,27)

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12002324, #12002329, #12002334, …
maximal subgroup of	C₅×#10351, C₅×#10352, C₅×#10353, …
maximal normal subgroup of	C₅×#10351, C₅×#10352, C₅×#10353, …
maximal quotient of	C₅×#13911, C₅×#13982, C₅×#14030, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18248);