C₅×#32804 #11955524

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×#32804

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

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

normal subgroups only

This group has 768 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) = 37
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 37 points:

(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,17,19)(16,18,20,21)(22,23)(24,27)(25,29)(26,28)(30,34)(31,35)(32,36)(33,37);(7,10)(8,12)(14,16)(15,18)(17,20)(19,21)(24,30)(25,32)(27,34)(29,36)(31,33)(35,37);(8,12)(11,13)(15,19)(18,21)(22,24,26,31)(23,27,28,35)(25,33,32,30)(29,37,36,34);(6,8,9,12)(7,11,10,13)(16,20)(18,21)(23,28)(27,35)(29,36)(34,37)

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, 25792);