C₅×#2863 #10616458

order	640
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#2863

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

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

normal subgroups only

This group has 408 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)(6,7,11,10)(8,13,9,12)(14,15,18,21)(16,19,20,17)(24,28)(26,29);(6,8,11,9)(7,12,10,13)(14,16,18,20)(15,19,21,17)(22,23,25,27)(24,26,28,29);(6,9,11,8)(7,13,10,12)(14,17,18,19)(15,20,21,16);(6,10,11,7)(8,13,9,12)(14,15)(16,19)(17,20)(18,21)(22,24)(23,26)(25,28)(27,29)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11987699, #11987719, #11987791, …
maximal subgroup of	C₅×#7669, C₅×#7673, C₅×#7689, …
maximal normal subgroup of	C₅×#7669, C₅×#7673, C₅×#7689, …
maximal quotient of	C₅×#14688, C₅×#14699, C₅×#14703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18540);