C₅×#7528 #11930248

order	1280
nilpotency class	6
isoclinism	isoclinic class of #7527
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7528

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1408
commutator	C₄×C₈	C₂×C₄×C₅
Frattini, cosocle	#492	C₂²×C₅
prime radical	C₅×#127	D₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#492 ◁ 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)(8,17,19,30)(9,21,23,13)(10,15,26,14)(11,16)(12,18,27,35)(20,34,25,33)(22,29)(24,31,37,32)(28,36);(6,8,11,20)(7,12,16,27)(9,22,24,36)(10,25,26,19)(13,30,32,33)(14,29,15,28)(17,21,34,31)(18,23,35,37)

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 516);