C₅×#33827 #11956547

order	1280
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#33827

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂³×C₅	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

464 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 77
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 77 points:

(1,2,3,4,5)(7,10)(8,12)(14,15,20,25)(16,21,27,42)(17,30,33,64)(18,34,38,63)(19,39,36,24)(22,45,47,31)(23,48,50,37)(26,52,53,75)(28,58,56,74)(29,61,57,44)(32,65,40,49)(35,51,66,46)(41,67,68,54)(43,71,70,60)(55,76,62,72)(59,73,77,69);(6,7)(8,11)(9,10)(12,13)(14,16,20,27)(15,21,25,42)(17,26,33,53)(18,28,38,56)(19,29,36,57)(22,41,47,68)(23,43,50,70)(24,44,39,61)(30,52,64,75)(31,54,45,67)(32,55,40,62)(34,58,63,74)(35,59,66,77)(37,60,48,71)(46,69,51,73)(49,72,65,76);(7,10)(11,13)(14,17,20,33)(15,22,25,47)(16,26,27,53)(18,35,38,66)(19,40,36,32)(21,41,42,68)(23,49,50,65)(24,51,39,46)(28,59,56,77)(29,62,57,55)(30,63,64,34)(31,48,45,37)(43,72,70,76)(44,73,61,69)(52,74,75,58)(54,71,67,60);(6,8,9,12)(7,11,10,13)(16,27)(21,42)(26,53)(28,56)(29,57)(41,68)(43,70)(44,61)(52,75)(54,67)(55,62)(58,74)(59,77)(60,71)(69,73)(72,76)

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, 26815);