C₅×#7076 #11929796

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#7076

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#327
commutator	C₂²×C₈	C₂×C₄×C₅
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₅
prime radical	C₅×#1999	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₂²⋊C₈ ◁ G

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

normal subgroups only

This group has 338 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) = 29
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15)(16,24,27,19)(17,21)(18,23)(20,25,28,29)(22,26);(7,10)(11,12)(15,19)(16,25)(17,26)(20,22)(21,28)(23,24)(27,29)

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, 64);