C₅×#12397 #11935117

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12397

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×D₄ ◁ G

278 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 82 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)(6,7,11,15)(8,16,19,20)(9,14,10,13)(12,18,21,17)(22,23,28,32)(24,33,36,51)(25,30,27,42)(26,38,41,52)(29,44,47,35)(31,37,49,40)(34,50,43,48)(39,45,53,46);(6,8,11,19)(7,12,15,21)(9,18,10,17)(13,20,14,16)(22,24)(23,29)(25,34)(26,39)(27,43)(28,36)(30,45)(31,48)(32,47)(33,38)(35,40)(37,44)(41,53)(42,46)(49,50)(51,52);(7,13)(9,10)(12,20)(14,15)(16,21)(17,18)(22,25,28,27)(23,30,32,42)(24,34,36,43)(26,37,41,40)(29,45,47,46)(31,38,49,52)(33,50,51,48)(35,39,44,53)

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, 5385);