#103869

order	512
nilpotency class	3
isoclinism	isoclinic class of #103845
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

487 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

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, 11065);