#119071

order	512
nilpotency class	3
isoclinism	isoclinic class of #119068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

511 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26267);