#146562

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10730
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

945 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dhi,f^b=fi,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53758);