#102465

order	512
nilpotency class	3
isoclinism	isoclinic class of #102429
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	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

540 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(2,6)(3,4)(7,8)(9,10)(11,18)(12,23)(13,16)(14,17)(15,20)(19,22)(21,24)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,17,24)(12,21,13,20)(16,19,23,18);(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,23)(14,17)(15,20)(18,22)(21,24)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,e^c=ei,f^c=fg

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, 9661);