C₅×#7448 #11930168

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7448

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1361
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₄×C₁₆	C₂²×C₅
prime radical	C₅×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

146 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 50 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)(6,7,9,15,25,18,27,34,12,19,26,17,11,28,10,16)(8,20,22,14,33,32,37,31,24,35,36,30,23,13,21,29);(7,13)(8,21)(9,10)(11,25)(14,17)(15,30)(16,29)(18,35)(19,32)(20,28)(22,23)(24,37)(26,27)(31,34)(33,36)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 436);