C₅×#11536 #11934256

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11536

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅
prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×Q₈ ◁ G

376 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,7,9,11,10,12,8,13)(14,15)(16,24,27,39)(17,21)(18,31,33,22)(19,23)(20,26)(25,40,28,42)(29,44,45,37)(30,38)(32,35)(34,41)(36,43);(14,16,19,27)(15,20,23,35)(17,28,30,25)(18,32,33,26)(21,36,38,34)(22,39,31,24)(29,41,45,43)(37,40,44,42);(6,8,10,9)(7,11,12,13)(14,17,19,30)(15,21,23,38)(16,25,27,28)(18,29,33,45)(20,34,35,36)(22,37,31,44)(24,40,39,42)(26,41,32,43)

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, 4524);