C₅×#12799 #11935519

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#12799

Subgroups

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

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

normal subgroups only

This group has 588 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) = 25
transitive degree	μ_t(G) = 160

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

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5787);