C₅×#7481 #11930201

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7481
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#7481

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1390
commutator	C₂×Q₈	C₄²×C₅
Frattini, cosocle	C₂×#124	C₂²×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#124 ◁ G

284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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)(8,16,18,32)(9,21,22,31)(10,24,26,15)(11,14)(12,17,27,35)(13,23)(19,34,20,33)(25,30)(28,36,29,37);(6,8)(7,12,14,28)(9,17)(10,18)(11,20)(13,16,30,33)(15,27,24,29)(19,26)(21,32,31,34)(22,36)(23,37)(25,35)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 469);