#102764

order	512
nilpotency class	3
isoclinism	isoclinic class of #102749
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	#32806	C₂
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

515 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 74 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)(10,14)(11,12)(15,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,32)(29,31);(2,5)(3,8)(6,7)(9,10)(11,16)(12,14)(13,15)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,16)(13,15)(20,30)(22,31)(23,28)(26,29)

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,c^b=cf,d^b=dg,f^b=fghi,d^c=dh,e^c=ei,f^c=fghi

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, 9960);