C₅×#18999 #11941719

order	1280
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#18999

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

850 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 320

A minimal faithful permutation representation — 4 generators on 53 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 11987);