#153339

order	512
nilpotency class	5
isoclinism	isoclinic class of #153131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2199
commutator, Frattini, prime radical, cosocle	#532	C₂³
socle	C₂	#13463
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

268 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60535);