#145514

order	512
nilpotency class	4
isoclinism	isoclinic class of #145508
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33811	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 315 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52710);