C₅×#2852 #10616447

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

Aliases

direct product

C₅×#2852

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

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

normal subgroups only

This group has 582 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) = 25
transitive degree	μ_t(G) = 320

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

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12271983, #12272014, #12272048, …
maximal subgroup of	C₅×#34990, C₅×#34991, C₅×#34995, …
maximal normal subgroup of	C₅×#34990, C₅×#34991, C₅×#34995, …
maximal quotient of	C₅×#14683, C₅×#14695, C₅×#14696, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18529);