C₅×#2760 #10616355

order	640
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#2760

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₅×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

520 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 176 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)(7,10)(8,12)(14,15,18,16)(17,20,21,19)(22,23)(24,28)(25,26)(27,29);(7,10)(11,13)(15,16)(17,21)(23,26)(24,29);(6,7,9,10)(8,11,12,13)(14,16,18,15)(17,20,21,19)(22,24)(23,27)(25,29)(26,28);(6,8)(7,11)(9,12)(10,13)(14,17)(15,19)(16,20)(18,21)(22,23)(24,27)(25,26)(28,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12011913, #12012397, #12307622, …
maximal subgroup of	C₅×#12187, C₅×#12263, C₅×#36776, …
maximal normal subgroup of	C₅×#12187, C₅×#12263, C₅×#36776, …
maximal quotient of	C₅×#14375, C₅×#14508, C₅×#14572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18437);