C₅×#8699 #11931419

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8699

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

598 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 37
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1687);