C₅×#7066 #11929786

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7066

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂²×C₄	C₂×C₅×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₅
socle	C₂²×C₅	#324
prime radical	C₂³×C₅	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₄⋊C₈ ◁ G

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 45 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 54);