C₅×#7424 #11930144

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7424

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1347
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₂×M₃₂	C₂²×C₅
prime radical	C₅×#2218	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂ ◁ G

146 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 46 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)(6,7,9,15,12,18,25,17)(8,13,21,29,24,31,23,36)(10,27,20,16,11,28,26,35)(14,32,30,22,33,37,34,19);(7,13)(8,19)(9,25)(10,26)(11,20)(14,16)(15,29)(17,36)(18,31)(21,37)(22,24)(23,32)(27,34)(28,30)(33,35)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 412);