C₅×#32868 #11955588

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#32868

Subgroups

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

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

normal subgroups only

This group has 596 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)(6,7,9,12)(8,11,13,10)(14,15)(16,19)(17,18)(20,21)(22,23)(24,27)(25,28)(26,29)(30,37)(31,35)(32,34)(33,36);(6,8)(7,10)(9,13)(11,12)(15,18)(19,21)(22,24,25,31)(23,27,28,35)(26,30,33,32)(29,37,36,34);(6,8)(7,11)(9,13)(10,12)(15,18)(16,20)(23,28)(24,30)(26,33)(27,34)(31,32)(35,37);(8,13)(10,11)(14,16,17,20)(15,19,18,21)(23,28)(24,31)(29,36)(30,32)

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, 25856);