C₅×#32716 #11955436

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×#32716

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₅×#2615	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₅×#41 ◁ G

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

normal subgroups only

This group has 906 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)(9,12)(10,13)(14,15)(16,18)(17,19)(20,21)(22,23,26,30)(24,31,33,28)(25,35,27,29)(32,37,34,36);(6,8)(7,10)(9,13)(11,12)(14,16)(15,18)(17,20)(19,21)(22,24,26,33)(23,28,30,31)(25,34,27,32)(29,36,35,37);(6,8)(7,11)(9,13)(10,12)(15,19)(18,21)(24,32)(25,27)(28,36)(29,35)(31,37)(33,34);(8,13)(10,11)(16,20)(18,21)(22,25,26,27)(23,29,30,35)(24,32,33,34)(28,37,31,36)

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, 25704);