#144468

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

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

533 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=di,b^2=d,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	#72227645
maximal normal subgroup of	#72227645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51664);