C₅×#7359 #11930079

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7359

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#354
commutator	C₄²	C₂×C₅×C₈
Frattini, cosocle	#382	C₂²×C₅
prime radical	C₂³×C₅	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#382 ◁ G

176 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 50 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)(6,7,8,10,9,11,12,13)(14,15)(16,24,26,39)(17,28,30,44)(18,32,34,22)(19,23)(20,25,35,42)(21,31)(27,41,33,40)(29,36,43,37)(38,45);(14,16)(15,20,23,37)(17,29)(18,33)(19,27)(21,24,38,41)(22,35,32,36)(25,30)(26,34)(28,39,44,40)(31,43)(42,45)

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, 347);