#104256

order	512
nilpotency class	3
isoclinism	isoclinic class of #104245
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
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

611 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,5)(4,7)(6,8)(13,16)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(2,5)(3,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,27,32,31);(1,2,4,7)(3,6,8,5)(9,11)(10,13)(12,15)(14,16)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fg

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, 11452);