#153233

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, 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

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

normal subgroups only

This group has 349 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,8,11,7,12,5,26)(3,15,17,9,19,41,27,36)(4,21,28,42,25,10,23,38)(6,14,32,44,31,13,29,43)(16,48,49,37,22,56,52,55)(18,47,53,35,20,46,54,34)(24,39,58,62,30,40,57,59)(33,51,45,63,61,50,60,64);(1,3,8,17,7,19,5,27)(2,9,11,41,12,36,26,15)(4,22,28,52,25,16,23,49)(6,18,32,53,31,20,29,54)(10,37,38,56,21,55,42,48)(13,34,43,47,14,35,44,46)(24,51,58,63,30,50,57,64)(33,59,45,39,61,62,60,40);(1,4,7,25)(2,10,12,21)(3,16,19,22)(5,28,8,23)(6,30,31,24)(9,33,36,61)(11,42,26,38)(13,39,14,40)(15,45,41,60)(17,52,27,49)(18,50,20,51)(29,57,32,58)(34,55,35,37)(43,59,44,62)(46,48,47,56)(53,63,54,64)

Polycyclic presentation:

2^9|a^2=dh,b^2=dh,c^2=i,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, 60429);