#140715

order	512
nilpotency class	3
isoclinism	isoclinic class of #9781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

303 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=di,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47911);