C₅×#12294 #11935014

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12294

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3259	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#424 ◁ G

302 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 640

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

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5282);