C₅×#7415 #11930135

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7414
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7415

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1345
commutator	C₂×M₁₆	C₂×C₄×C₅
Frattini, cosocle	#423	C₂²×C₅
prime radical	C₅×#575	C₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#423 ◁ G

190 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 37 points:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 403);