C₅×#7190 #11929910

order	1280
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7190

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂×C₄	C₄×C₅×C₈
Frattini, cosocle	C₂×#117	C₂²×C₅
socle	C₂²×C₅	#342
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

322 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,7,8,10,9,11,12,13)(14,15)(16,24,26,36)(17,28,29,21)(18,31)(19,23)(20,25)(22,32)(27,41,40,37)(30,35,43,42)(33,45)(34,39)(38,44);(14,16,18,27,19,26,32,40)(15,20,22,33,23,34,31,44)(17,25,30,38,29,39,43,45)(21,24,35,37,28,36,42,41)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 178);