C₅×#13855 #11936575

order	1280
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#13855

Subgroups

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

1246 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 494 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) = 33
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 4 generators on 33 points:

(1,2,3,4,5)(6,7,8,9)(10,11)(12,17)(13,15)(14,16)(18,19)(20,22)(21,23)(24,25);(10,12,14,13)(11,15,16,17)(18,20,21,24)(19,22,23,25)(26,27)(28,31)(29,30)(32,33);(10,13,14,12)(11,15,16,17)(19,23)(22,25)(27,30)(31,33);(26,28,29,32)(27,31,30,33)

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6843);