C₅²×#416 #13103153

order	1600
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅² · Schur multiplier (H₂) = C₂³×C₅

Aliases

direct product

C₅²×#416

Subgroups

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

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

normal subgroups only

This group has 632 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)(19,27)(20,23)(21,25)(26,29)(28,30);(6,7,8,9,10)(15,17)(16,21)(18,25)(19,28)(20,26)(22,29)(23,30)(24,27);(15,18,20,19)(16,22,23,27)(17,25,26,28)(21,29,30,24)

Conjugacy classes & character table

700 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5505);