C₅²×#405 #13103142

order	1600
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅² · Schur multiplier (H₂) = C₂×C₄×C₅

Aliases

direct product

C₅²×#405

Subgroups

	subgroup	G/H
center	C₂×C₅²×C₈	C₂²
commutator	C₂	C₂×C₄²×C₅²
Frattini, cosocle	C₂×C₄	C₂³×C₅²
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅²	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅²×C₈ ◁ G

520 conjugacy classes of subgroups; only the 456 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 456 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)(15,16,18,21,19,22,25,28)(17,20,24,27,23,26,29,30);(6,7,8,9,10)(11,12,13,14)(15,17)(16,20)(18,24)(19,23)(21,27)(22,26)(25,29)(28,30);(17,23)(20,26)(24,29)(27,30)

Conjugacy classes & character table

1000 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5494);