C₅×#8520 #11931240

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#8520

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

976 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

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

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, 1508);