C₅×#12892 #11935612

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#12892

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

274 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 158 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 — 3 generators on 45 points:

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5880);