C₅×#7224 #11929944

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7224

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#340
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
prime radical	C₂³×C₅	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 53
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2,3,4,5)(6,7)(8,14,16,20)(9,12)(10,13)(11,15,18,21)(17,19)(22,23,26,31,28,34,45,33)(24,35,37,48,40,50,39,53)(25,41,43,30,44,32,27,46)(29,38,47,36,49,52,51,42);(6,8)(7,11,13,18)(9,15)(10,16)(12,14,19,20)(17,21)(22,24,27,38,45,37,44,36,28,40,43,52,26,39,25,42)(23,29,32,48,31,51,46,35,34,49,41,53,33,47,30,50)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 212);