#145060

order	512
nilpotency class	4
isoclinism	isoclinic class of #143259
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1624
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10413
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 285 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52256);