C₅×#7287 #11930007

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#7287

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂×C₄×D₄	C₂²×C₅
socle	C₂²×C₅	#342
prime radical	C₂²×C₅×D₄	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×D₄ ◁ G

322 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 320

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

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 275);