C₅×#32857 #11955577

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#32857

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

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

normal subgroups only

This group has 510 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) = 1280

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

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

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, 25845);