#144458

order	512
nilpotency class	4
isoclinism	isoclinic class of #142258
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

508 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,e^2=gi,f^2=i,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=fi,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51654);