#143658

order	512
nilpotency class	4
isoclinism	isoclinic class of #143657
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33771	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 430 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=gi,e^2=hi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fghi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fghi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50854);