C₅×#7411 #11930131

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7410
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7411

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1344
commutator	C₂²×C₈	C₂×C₄×C₅
Frattini, cosocle	#408	C₂²×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#408 ◁ G

162 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 399);