C₅×#2825 #10616420

order	640
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₅×#2825

Subgroups

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

522 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 168 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)(11,13)(14,15)(16,19)(17,18)(20,21)(22,23,26,27)(24,28,29,25);(8,12)(11,13)(15,18)(16,20)(24,29)(25,28);(6,7)(8,11)(9,10)(12,13)(14,15)(16,21)(17,18)(19,20)(22,24)(23,25)(26,29)(27,28);(6,8)(7,11)(9,12)(10,13)(14,16)(15,19)(17,20)(18,21)(22,25)(23,24)(26,28)(27,29)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#2825, #12305695, #12305737, …
maximal subgroup of	C₂×C₅×#2825, C₅×#36670, C₅×#36676, …
maximal normal subgroup of	C₂×C₅×#2825, C₅×#36670, C₅×#36676, …
maximal quotient of	C₅×#14621, C₅×#14637, C₅×#14642, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18502);