C₅×#7234 #11929954

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7234

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#340
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂³×C₈	C₂²×C₅
prime radical	C₂⁴×C₅	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

264 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 320

A minimal faithful permutation representation — 2 generators on 45 points:

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

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, 222);