#104725

order	512
nilpotency class	3
isoclinism	isoclinic class of #102905
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

921 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 921 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=dgi,b^2=dg,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72238087
maximal subgroup of	#72238087
maximal normal subgroup of	#72238087
maximal quotient of	#72238061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11921);