#142144

order	512
nilpotency class	4
isoclinism	isoclinic class of #125524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

255 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,19,14,18,32,36,15,21,33,28,34,20,38,23)(11,22,25,39,26,12,17,30,27,16,35,37,24,29,31,40);(2,5)(6,7)(9,11,13,31,14,24,32,35,15,27,33,17,34,26,38,25)(10,16,36,39,20,29,19,30,21,22,23,37,18,12,28,40);(1,3,4,8)(2,6,5,7)(9,12,14,16,15,29,34,22)(10,17,18,35,21,31,20,25)(11,23,26,19,27,36,24,28)(13,30,32,37,33,40,38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eh,d^2=ghi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,f^b=fhi,h^b=hi,d^c=di

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49340);