C₅×#7413 #11930133

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×#7413

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

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

normal subgroups only

This group has 178 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)(6,7)(8,10)(9,12)(11,13)(14,15,17,22,19,24,30,39)(16,25,27,42,26,40,43,38)(18,21,33,37,34,23,31,41)(20,29,35,32,36,44,45,28);(7,10)(8,13)(11,12)(14,16)(15,20,24,36)(17,29)(18,32)(19,26)(21,25,23,40)(22,38,39,42)(27,33)(28,34)(30,44)(31,43)(35,41,45,37)

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, 401);