#149396

order	512
nilpotency class	4
isoclinism	isoclinic class of #13522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2088
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11378
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

439 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:

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

Polycyclic presentation:

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

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, 56592);