C₅×#7059 #11929779

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

Aliases

direct product

C₅×#7059

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂²×C₄	C₂×C₅×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₅
socle	C₂²×C₅	#323
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₂²⋊C₈ ◁ G

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

normal subgroups only

This group has 194 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2,3,4,5)(6,7)(8,14)(9,17,18,12)(10,13)(11,15,19,21)(16,20)(22,23,26,31,27,32,41,49,28,33,42,50,25,30,39,47)(24,34,36,51,35,45,52,48,37,46,40,53,38,29,43,44);(6,8,10,16)(7,11,13,19)(9,15,18,21)(12,14,17,20)(22,24)(23,29)(25,38)(26,40)(27,35)(28,37)(30,46)(31,48)(32,34)(33,45)(36,42)(39,52)(41,43)(44,50)(47,51)(49,53)

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, 47);