C₅×#12628 #11935348

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₅×#12628

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂²×D₄	C₂³×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×D₄ ◁ G

628 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 25
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 25 points:

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

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, 5616);