C₅×#32794 #11955514

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#32794

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#117 ◁ G

600 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 206 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)(6,7)(8,10)(9,11)(12,13)(14,15,19,22)(16,23,17,20)(18,21,26,29)(24,28,25,27)(30,31)(32,35)(33,37)(34,36)(38,42)(39,44)(40,43)(41,45);(6,8)(7,10)(9,12)(11,13)(14,16,19,17)(15,20,22,23)(18,25,26,24)(21,28,29,27)(30,32,34,40)(31,35,36,43)(33,38,41,39)(37,42,45,44);(7,11)(10,13)(14,17,19,16)(15,20,22,23)(18,24,26,25)(21,28,29,27)(32,38)(33,41)(35,42)(37,45)(39,40)(43,44);(8,12)(10,13)(14,18,19,26)(15,21,22,29)(16,24,17,25)(20,27,23,28)(31,36)(35,43)(37,45)(42,44)

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, 25782);