#102806

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

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

normal subgroups only

This group has 643 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)(3,5)(4,6)(7,8)(9,10,12,11)(13,14,16,15)(17,18)(19,26)(20,22)(21,31)(23,25)(24,27)(28,32)(29,30);(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,28)(18,24,20,30)(21,29,23,27)(25,32,31,26);(1,3)(2,5)(4,7)(6,8)(10,14)(11,15)(13,16)(17,20,22,18)(19,26,28,32)(21,31,23,25)(24,27,30,29)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 10002);