#104936

order	512
nilpotency class	3
isoclinism	isoclinic class of #104917
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

322 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,7)(4,6)(5,8)(10,11)(12,13)(15,16)(17,18,20,24)(19,26,27,22)(21,23,29,25)(28,32,30,31);(1,2,3,7)(4,8,5,6)(9,10)(11,13)(12,15)(14,16)(17,19,20,27)(18,22,24,26)(21,30,29,28)(23,32,25,31);(2,6)(4,5)(7,8)(9,11,13,10,14,15,12,16)(18,23)(21,29)(22,31)(24,25)(26,32)(28,30)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=dh,d^2=g,e^2=h,f^2=ghi,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=fghi,e^c=egh,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12132);