C₅×#9516 #11932236

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9516

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂×C₅×#120	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

626 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 122 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)(6,7,11,12,10,13,9,8)(14,15)(16,19,23,27)(17,24,25,20)(18,21)(22,26)(28,29);(6,8,11,13,10,12,9,7)(14,16)(15,19,21,27)(17,22)(18,23)(20,26,24,29)(25,28);(6,9,10,11)(7,12,13,8)(14,17,18,25)(15,20,21,24)(16,22,23,28)(19,26,27,29)

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, 2504);