#105166

order	512
nilpotency class	3
isoclinism	isoclinic class of #105137
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#394
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

316 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=dh,d^2=g,e^2=h,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,e^c=egh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12362);