#153296

order	512
nilpotency class	5
isoclinism	isoclinic class of #153295
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13422
commutator, Frattini, cosocle	#532	C₂³
prime radical	#3018	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

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

normal subgroups only

This group has 385 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,d^b=di,e^b=eg,f^b=fi,g^b=gi,h^b=hi,e^c=ehi,g^c=gh,f^e=fh,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, 60492);