C₅×#13055 #11935775

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13055

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×Q₈	C₂³×C₅
prime radical	C₂×C₅×#454	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×Q₈ ◁ G

680 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 320

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

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

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, 6043);