#148295

order	512
nilpotency class	4
isoclinism	isoclinic class of #13400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#11095
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

647 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,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, 55491);