#148513

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2054
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

635 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55709);