C₅×#8309 #11931029

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#8309

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁵×C₅	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

874 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7)(8,9)(10,11)(12,18,20,15)(13,16)(14,17)(19,24,25,22)(21,23)(26,27,29,33,30,32,38,40)(28,34,36,41,35,31,37,39);(10,12)(11,15)(13,19)(14,20)(16,22)(17,18)(21,25)(23,24)(26,28)(27,31)(29,37)(30,35)(32,34)(33,41)(36,38)(39,40);(7,9)(10,13,14,21)(11,16,17,23)(12,19,20,25)(15,22,18,24)(28,35)(31,34)(36,37)(39,41)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1297);