#153232

order	512
nilpotency class	5
isoclinism	isoclinic class of #153232
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13377
commutator, Frattini, cosocle	#532	C₂³
prime radical	#3549	C₂²
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=dh,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fgh,h^a=hi,c^b=cf,d^b=di,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=di,g^c=gh,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60428);