C₅×#7233 #11929953

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7233

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#340
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂³×C₈	C₂²×C₅
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

308 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 320

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

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 221);