#141459

order	512
nilpotency class	4
isoclinism	isoclinic class of #141451
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

373 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5,7,8)(4,6)(9,10,15,21,16,23,33,22)(11,24,27,40,28,18,38,39)(12,29,26,37,31,36,25,17)(13,32,30,19,14,34,35,20);(1,3,4,7)(2,5)(6,8)(9,11,13,25,33,38,35,31,16,28,14,26,15,27,30,12)(10,17,20,18,21,29,32,39,23,37,19,24,22,36,34,40);(2,6)(5,8)(9,12,30,27,15,26,14,28,16,31,35,38,33,25,13,11)(10,18,32,37,22,40,20,29,23,24,34,17,21,39,19,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=egi,d^2=ghi,e^2=ghi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48655);