C₅×#8244 #11930964

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#8244

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂³×C₅	#117
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

368 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 170 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)(6,7,10,14,11,15,9,12)(8,13,18,20,16,21,17,19)(22,23,25,28)(24,30,31,27)(26,34,33,29)(32,35,37,36);(7,12,15,14)(8,16)(13,20,21,19)(17,18)(22,24,26,32,25,31,33,37)(23,27,29,35,28,30,34,36);(6,8,10,18,11,16,9,17)(7,13,14,20,15,21,12,19)

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, 1232);