#99710

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 536 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gh,d^2=g,e^2=hi,f^2=ghi,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6906);