#102800

order	512
nilpotency class	3
isoclinism	isoclinic class of #102785
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

543 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=gi,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=fgi,d^c=dh,e^c=ei,f^c=fgi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9996);