#145540

order	512
nilpotency class	4
isoclinism	isoclinic class of #145540
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10536
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33717	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 353 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) = 32

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

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

Polycyclic presentation:

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

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, 52736);