#105171

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, prime radical	C₂⁴	#120
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

675 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12367);