C₂×C₅×#1407 #11936296

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×C₅×#1407

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#353
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	#124	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#124 ◁ G

416 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 39
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2)(35,36,37,38,39);(3,4)(5,13,15,28)(6,17,18,33)(7,20,22,11)(8,12)(9,14,23,30)(10,19)(16,26,21,29)(24,31,25,32)(27,34);(3,5,8,16)(4,9,12,25)(6,14,18,32)(7,21,22,15)(10,26,27,13)(11,23,20,24)(17,29,33,28)(19,30,34,31)

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, 6564);