#102644

order	512
nilpotency class	3
isoclinism	isoclinic class of #102577
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	C₂⁴	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

683 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 683 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9840);