C₅×#2327 #10615922

order	640
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#2327

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂⁴
commutator	C₂²	C₂³×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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)(7,9)(10,11,13,12)(15,19)(16,22)(17,25)(27,29);(10,12,13,11)(14,15)(16,23)(17,21)(18,19)(20,22)(24,29)(25,26)(27,28);(6,7,8,9)(14,16)(15,20)(17,24)(18,22)(19,23)(21,27)(25,28)(26,29);(14,17,18,25)(15,21,19,26)(16,24,22,28)(20,27,23,29)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11994801, #11994821, #11996006, …
maximal subgroup of	C₅×#9100, C₅×#9104, C₅×#9296, …
maximal normal subgroup of	C₅×#9100, C₅×#9104, C₅×#9296, …
maximal quotient of	C₅×#13779, C₅×#13786, C₅×#13790, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18004);