C₅²×#419 #13103156

order	1600
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅² · Schur multiplier (H₂) = C₂³×C₅

Aliases

direct product

C₅²×#419

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅²	C₂×D₄
commutator	C₄	C₂²×C₄×C₅²
Frattini, cosocle	C₂×C₄	C₂³×C₅²
prime radical	C₂³×C₅²	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₂²×C₄×C₅² ◁ G

600 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 328 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) = 30
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

550 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5508);