C₅×#7038 #11929758

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7038

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#95
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂²×C₄²	C₂²×C₅
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

358 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 29
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 26);