C₅×#32768 #11955488

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₅×#32768

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#117 ◁ G

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

normal subgroups only

This group has 736 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) = 640

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)(16,18)(17,19)(20,21)(22,23)(24,28)(25,29)(26,31)(27,30)(32,41)(33,43)(34,42)(35,44)(36,46)(37,45)(38,48)(39,49)(40,47)(50,52)(51,53);(7,10)(8,12)(14,16)(15,18)(17,20)(19,21)(22,24,27,34)(23,28,30,42)(25,35,37,51)(26,38,40,33)(29,44,45,53)(31,48,47,43)(32,39,50,36)(41,49,52,46);(8,12)(11,13)(15,19)(18,21)(22,25,27,37)(23,29,30,45)(24,32,34,50)(26,39,40,36)(28,41,42,52)(31,49,47,46)(33,51,38,35)(43,53,48,44);(6,8,9,12)(7,11,10,13)(16,20)(18,21)(23,30)(28,42)(29,45)(31,47)(41,52)(43,48)(44,53)(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, 25756);