#100786

order	512
nilpotency class	3
isoclinism	isoclinic class of #100723
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

559 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=gh,e^2=hi,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982);