#144535

order	512
nilpotency class	4
isoclinism	isoclinic class of #144523
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

303 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=h,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=egi,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51731);