C₅×#12970 #11935690

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12970

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	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×Q₈ ◁ G

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

normal subgroups only

This group has 472 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)(6,7,9,8)(10,11)(12,15)(13,16)(14,19)(17,20)(18,22)(21,24)(23,25);(6,8,9,7)(11,15)(12,20)(13,22)(16,24)(17,19)(18,25);(11,16)(13,22)(15,24)(17,25)(18,19)(21,23)

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, 5958);