C₅×#8521 #11931241

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#8521

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

690 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1509);