#144507

order	512
nilpotency class	4
isoclinism	isoclinic class of #142220
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26,28,30)(27,31,34,38)(29,36,35,32)(33,40,37,39);(1,3,4,8)(2,5,6,7)(10,14)(15,23)(16,21)(17,24)(26,30)(27,33)(29,35)(31,39)(34,37)(38,40);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(25,27)(26,31)(28,34)(29,37)(30,38)(32,39)(33,35)(36,40)

Polycyclic presentation:

2^9|a^2=d,b^2=dgi,c^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi,h^c=hi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266618
maximal normal subgroup of	#72266618
maximal quotient of	#72266620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51703);