C₅×#7521 #11930241

order	1280
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7521

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1402
commutator	C₄⋊Q₈	C₂×C₄×C₅
Frattini, cosocle	#494	C₂²×C₅
prime radical	C₅×#555	C₄
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#494 ◁ G

148 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 30 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)(6,7,9,14,10,16,22,34)(8,17,19,31,18,28,37,32)(11,15,26,35,24,36,23,33)(12,21,27,25,13,29,30,20);(7,12)(8,18)(13,16)(14,31)(15,17)(19,37)(20,25)(21,29)(27,33)(28,36)(30,35)(32,34)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 509);