C₅×#2621 #10616216

order	640
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#2621

Subgroups

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

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

normal subgroups only

This group has 528 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) = 320

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

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11995324, #11995329, #12009489, …
maximal subgroup of	C₅×#9195, C₅×#9196, C₅×#11694, …
maximal normal subgroup of	C₅×#9195, C₅×#9196, C₅×#11694, …
maximal quotient of	C₅×#14258, C₅×#14261, C₅×#14274, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18298);