#149130

order	512
nilpotency class	4
isoclinism	isoclinic class of #140203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1934
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
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

417 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)(3,17)(4,23,28,54)(5,30)(6,12)(7,13)(8,35)(9,16)(10,19)(11,26,44,60)(14,34)(15,32)(18,49,51,37)(20,47)(21,38)(22,48)(24,55,57,64)(25,58,27,45)(29,43,59,42)(31,39)(33,40)(36,46)(41,50,63,61)(52,62,56,53);(1,3)(2,10,16,40)(4,24)(5,31)(6,33)(7,20)(8,36)(9,22)(11,41,44,63)(12,38,14,46)(13,17,30,48)(15,47,35,39)(18,27)(19,34)(21,32)(23,53,54,62)(25,51)(26,52)(28,57)(29,61)(37,42,49,43)(45,55,58,64)(50,59)(56,60);(1,4,5,25,32,59,34,60,9,28,7,27,8,29,6,26)(2,11,12,42,35,58,13,23,16,44,14,43,15,45,30,54)(3,18,19,24,36,56,31,61,22,51,33,57,21,52,20,50)(10,37,17,41,47,53,38,55,40,49,48,63,39,62,46,64)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,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, 56326);