#153338

order	512
nilpotency class	5
isoclinism	isoclinic class of #153125
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

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

normal subgroups only

This group has 271 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=dh,c^2=d,d^2=i,e^2=i,f^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,f^b=fi,g^b=gi,h^b=hi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60534);