C₅×#7335 #11930055

order	1280
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7335

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₅×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅
socle	C₂²×C₅	C₂²⋊C₁₆
prime radical	C₂×C₅×#41	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

158 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 2 generators on 77 points:

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

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 323);