C₅²×#410 #13103147

order	1600
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅² · Schur multiplier (H₂) = C₂²×C₄×C₅

Aliases

direct product

C₅²×#410

Subgroups

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

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

normal subgroups only

This group has 632 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)(13,19,21,16)(14,17)(15,18)(20,25,26,23)(22,24);(6,7,8,9,10)(11,13)(12,16)(14,20)(15,21)(17,23)(18,19)(22,26)(24,25);(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)

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, 5499);