C₂×C₅×#1323 #11934162

order	1280
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₅×#1323

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	C₂²⋊C₈
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

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

normal subgroups only

This group has 394 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) = 47
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2)(43,44,45,46,47);(3,4)(5,9,10,7)(6,8)(11,12,15,20,16,21,30,38,17,22,31,39,14,19,28,36)(13,23,25,40,24,34,41,37,26,35,29,42,27,18,32,33);(3,5,6,10)(4,7,8,9)(11,13)(12,18)(14,27)(15,29)(16,24)(17,26)(19,35)(20,37)(21,23)(22,34)(25,31)(28,41)(30,32)(33,39)(36,40)(38,42)

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4430);