#146440

order	512
nilpotency class	4
isoclinism	isoclinic class of #12833
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1807
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10721
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

861 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53636);