#144474

order	512
nilpotency class	4
isoclinism	isoclinic class of #142216
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×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 651 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)(17,18,21,25)(19,22,27,24)(20,28)(23,29)(26,32)(30,31);(1,3,4,8)(2,5,6,7)(17,19)(18,22)(20,29,30,26)(21,27)(23,28,32,31)(24,25);(9,10,12,14)(11,16,15,13)(17,20,21,30)(18,23,25,32)(19,26,27,29)(22,31,24,28)

Polycyclic presentation:

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

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, 51670);