C₅×#12511 #11935231

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₅×#12511

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

260 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 53 points:

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

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5499);