C₅×#33825 #11956545

order	1280
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#33825

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

512 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 194 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)(16,20)(17,21)(18,31)(19,24)(22,28)(23,29)(25,33)(26,32)(27,41)(30,37)(34,40)(35,39)(36,42)(38,43)(44,45);(6,7)(8,11)(9,10)(12,13)(14,16,19,25)(15,20,24,33)(17,26,30,38)(18,27,29,39)(21,32,37,43)(22,34,36,44)(23,35,31,41)(28,40,42,45);(7,10)(11,13)(15,21)(17,28)(18,29)(20,32)(22,23)(24,37)(26,40)(27,39)(30,42)(31,36)(33,43)(34,35)(38,45)(41,44);(6,8,9,12)(7,11,10,13)(16,25)(20,33)(26,38)(27,39)(32,43)(34,44)(35,41)(40,45)

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, 26813);