#149362

order	512
nilpotency class	4
isoclinism	isoclinic class of #135291
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1825
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11372
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

431 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,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, 56558);