C₅×#11859 #11934579

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11859

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

314 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7)(8,14,16,11)(9,12)(10,13)(15,20,17,19)(18,21)(22,23)(24,34)(25,31)(26,41)(27,43)(28,33)(29,36)(30,39)(32,42)(35,51)(37,45)(38,48)(40,50)(44,49)(46,53)(47,52);(6,8)(7,11)(9,17)(10,16)(12,20)(13,14)(15,18)(19,21)(22,24,28,38)(23,29,33,47)(25,36,40,52)(26,35,39,53)(27,44,42,37)(30,48,41,34)(31,49,50,45)(32,51,43,46);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,25,27,39,28,40,42,26)(23,30,32,31,33,41,43,50)(24,35,37,52,38,53,44,36)(29,45,46,34,47,49,51,48)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4847);