#144748

order	512
nilpotency class	4
isoclinism	isoclinic class of #144720
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=d,e^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fi,e^c=eg,f^c=fi,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, 51944);