C₅×#7419 #11930139

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7419

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#357
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅
prime radical	C₂³×C₅	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

152 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(15,21,22,37)(16,26)(17,32)(18,28)(19,34)(23,40,24,41)(25,42,36,39)(27,33,38,35)(29,31)(30,45)(43,44);(6,8)(7,10)(9,13)(11,12)(14,15,20,22)(16,27,31,38)(17,33,18,35)(19,36,34,25)(21,26,37,29)(23,28,24,32)(30,40,43,41)(39,45,42,44)

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 407);