C₅×#11504 #11934224

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11504

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

510 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,7,8,11,9,10,12,13)(14,15)(16,24)(17,21)(18,31,32,22)(19,23)(20,26,34,41)(25,38)(27,39)(28,42)(29,44,45,36)(30,37)(33,40,35,43);(14,16,19,27)(15,20,23,34)(17,28,30,25)(18,26,32,41)(21,35,37,33)(22,24,31,39)(29,43,45,40)(36,42,44,38);(7,10)(11,13)(14,17,19,30)(15,21,23,37)(16,25,27,28)(18,29,32,45)(20,33,34,35)(22,36,31,44)(24,38,39,42)(26,40,41,43)

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, 4492);