C₅×#8515 #11931235

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8515

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#3567	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

644 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,10,17,13,8,19,23)(9,25,28,61,30,51,47,60)(11,33,31,43,36,59,48,54)(12,18,32,15,37,24,49,21)(14,38,40,65,20,50,39,27)(16,44,46,56,22,55,58,35)(26,62,29,45,64,69,66,57)(34,41,67,42,68,52,63,53);(6,8)(7,13)(9,20)(10,15)(11,33)(12,24)(14,30)(16,35)(17,32)(18,37)(19,21)(22,56)(23,49)(25,34)(26,39)(27,45)(28,42)(29,52)(31,58)(36,59)(38,60)(40,64)(41,66)(43,55)(44,54)(46,48)(47,53)(50,61)(51,68)(57,65)(62,67)(63,69);(6,9,12,29)(7,14,18,41)(8,20,24,52)(10,28,32,64)(11,34,35,27)(13,30,37,66)(15,42,17,40)(16,45,33,25)(19,47,49,26)(21,53,23,39)(22,57,59,51)(31,67,44,38)(36,68,56,65)(43,61,46,69)(48,63,55,50)(54,60,58,62)

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, 1503);