C₅×#2641 #10616236

order	640
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₅×#2641

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂×C₅×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

584 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12000649, #12000705, #12074412, …
maximal subgroup of	C₅×#10089, C₅×#10097, C₅×#18430, …
maximal normal subgroup of	C₅×#10089, C₅×#10097, C₅×#18430, …
maximal quotient of	C₅×#14260, C₅×#14272, C₅×#14283, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18318);