#148554

order	512
nilpotency class	4
isoclinism	isoclinic class of #140140
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1925
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 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)(3,14,18,46)(4,20)(5,26)(6,30)(7,13)(8,16,34,17)(9,22)(10,28)(11,41)(12,32)(15,47,49,62)(19,40,31,42)(21,45,48,36)(23,39)(24,59)(25,37)(27,44,51,35)(29,43)(33,54,61,50)(38,58)(52,57)(53,56)(55,63,60,64);(1,3)(2,8)(4,21)(5,27)(6,31)(7,18)(9,36)(10,40)(11,42)(12,44)(13,34)(14,43)(15,23)(16,41)(17,28)(19,32)(20,33)(22,49)(24,60)(25,48)(26,46)(29,51)(30,35)(37,61)(38,64)(39,45)(47,59)(50,57)(52,63)(53,62)(54,58)(55,56);(1,4,5,22,6,24,28,57,7,25,29,23,32,56,41,58)(2,9,10,37,12,38,26,59,13,39,11,20,30,52,43,53)(3,15,16,48,19,50,51,60,18,49,17,21,31,54,27,55)(8,33,14,45,35,47,42,64,34,61,46,36,44,62,40,63)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,g^b=gi,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, 55750);