#146538

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1823
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

941 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi,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, 53734);