C₅×#2225 #10603733

order	640
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#2225

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ G

130 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 3 generators on 29 points:

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

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12013086, #12013093, #12013142, …
maximal subgroup of	C₅×#12369, C₅×#12370, C₅×#12377, …
maximal normal subgroup of	C₅×#12369, C₅×#12370, C₅×#12377, …
maximal quotient of	C₅×#12812, C₅×#12938, C₅×#13110, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5815);