#149094

order	512
nilpotency class	4
isoclinism	isoclinic class of #13400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11293
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

455 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=gh,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, 56290);