C₅×#7522 #11930242

order	1280
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7522

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1402
commutator	C₄⋊Q₈	C₂×C₄×C₅
Frattini, cosocle	#494	C₂²×C₅
prime radical	C₂²×C₅	AΣL(1,𝔽₁₆)
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#494 ◁ G

142 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 30 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)(6,7,10,14,11,16,24,34)(8,15,18,35,19,30,25,33)(9,12,21,26,22,28,36,32)(13,29,31,20,27,23,37,17);(6,8,11,19)(7,12,16,28)(9,20,22,17)(10,18,24,25)(13,30,27,15)(14,31,34,37)(21,23,36,29)(26,35,32,33)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 510);