#153236

order	512
nilpotency class	5
isoclinism	isoclinic class of #153235
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, prime radical, cosocle	#532	C₂³
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

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

normal subgroups only

This group has 359 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=dh,c^2=i,d^2=i,f^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,g^b=gi,h^b=hi,d^c=di,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, 60432);