#149048

order	512
nilpotency class	4
isoclinism	isoclinic class of #140091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1918
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11292
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

449 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dh,f^b=fhi,g^b=ghi,h^b=hi,g^c=gi

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, 56244);