C₅×#8527 #11931247

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8527

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 656 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)(6,7,10,15,11,14,9,13)(8,16,17,21,18,12,19,20)(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);(6,8,11,18)(7,12,14,16)(9,17,10,19)(13,20,15,21)(22,24)(23,28)(25,33)(26,39)(27,35)(29,41)(30,44)(31,42)(32,45)(34,40)(36,50)(37,52)(38,49)(43,47)(46,51)(48,53);(6,9,11,10)(7,13,14,15)(8,17,18,19)(12,20,16,21)(22,25,27,38)(23,29,31,36)(24,33,35,49)(26,37,40,53)(28,41,42,50)(30,43,45,51)(32,46,44,47)(34,48,39,52)

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, 1515);