C₅×#9499 #11932219

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#9499

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂⁵	C₂³×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁵×C₅ ◁ G

956 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 160

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2487);