C₅×#32882 #11955602

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₅×#32882

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁵×C₅	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂⁴×C₄×C₅ ◁ G

3798 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 3798 conjugacy classes of subgroups — too many to draw. Only its 438 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) = 320

A minimal faithful permutation representation — 4 generators on 29 points:

(1,2,3,4,5)(6,7)(8,9)(10,11)(12,13)(14,15,18,21)(16,22,24,29)(17,25,27,20)(19,26,28,23);(14,16,18,24)(15,19,21,28)(17,26,27,23)(20,29,25,22);(7,9);(11,13)

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 25870);