#148303

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
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:

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

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,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=di,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, 55499);