C₅×#12631 #11935351

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#12631

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂²×Q₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×Q₈ ◁ G

464 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7,9,8)(10,11)(12,20)(13,24)(14,19)(15,17)(16,21)(18,26)(22,38)(23,34)(25,33)(27,32)(28,31)(29,37)(30,39)(35,40)(36,41);(10,12,14,22,15,23,28,40)(11,16,19,30,17,29,31,41)(13,21,25,36,26,37,27,39)(18,20,32,35,24,34,33,38);(6,8,9,7)(11,17)(16,29)(18,24)(19,31)(20,34)(30,41)(32,33)(35,38)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5619);