C₅×#12874 #11935594

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12874

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1823
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

376 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 5862);