C₅²×#421 #13103158

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×#421

Subgroups

	subgroup	G/H
center	C₄×C₅²	C₂×D₄
commutator	C₄	C₂²×C₄×C₅²
Frattini, cosocle	C₂×C₄	C₂³×C₅²
socle	C₂×C₅²	C₂×C₂²⋊C₄
prime radical	C₂×C₅²×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅² ◁ C₂²×C₄×C₅² ◁ G

616 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 312 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) = 26
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 3 generators on 26 points:

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

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, 5510);