#104749

order	512
nilpotency class	3
isoclinism	isoclinic class of #104729
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

629 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dgh,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,c^b=cf,e^b=ei,d^c=dg,e^c=egh,f^c=fg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11945);