#146439

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^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, 53635);