#102851

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

671 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,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, 10047);