C₅×#7429 #11930149

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7429

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#354
commutator	C₂×C₄²	C₂×C₄×C₅
Frattini, cosocle	#391	C₂²×C₅
prime radical	C₂³×C₅	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#391 ◁ G

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

normal subgroups only

This group has 196 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) = 45
transitive degree	μ_t(G) = 320

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

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

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, 417);