C₅×#33821 #11956541

order	1280
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#33821

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂³×C₅	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

484 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 640

A minimal faithful permutation representation — 4 generators on 45 points:

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

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, 26809);