C₅×#32816 #11955536

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₄³

Aliases

direct product

C₅×#32816

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

662 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 1280

A minimal faithful permutation representation — 4 generators on 53 points:

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

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, 25804);