C₅×#7077 #11929797

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7077

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#327
commutator	C₂²×C₈	C₂×C₄×C₅
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₅
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₂²⋊C₈ ◁ G

218 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 320

A minimal faithful permutation representation — 2 generators on 37 points:

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

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, 65);