#146610

order	512
nilpotency class	4
isoclinism	isoclinic class of #12767
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1790
commutator	C₂×D₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10745
prime radical	#3449	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

966 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)(3,13)(4,18,20,9)(5,22,23,10)(6,12)(7,11)(8,15,27,40)(14,34,38,50)(16,36)(17,35)(19,29)(21,44,47,30)(24,48,49,32)(25,33)(26,37)(28,41,52,60)(31,45)(39,55,59,63)(42,56)(43,58)(46,53)(51,57)(54,62)(61,64);(1,3,6,17,7,16,25,42)(2,8,12,28,11,27,33,52)(4,14,21,39,20,38,47,59)(5,15,24,41,23,40,49,60)(9,26,30,51,18,43,44,61)(10,13,32,35,22,36,48,56)(19,37,46,57,45,58,62,64)(29,50,53,63,31,34,54,55);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,26)(10,31)(11,18)(12,30)(13,34)(15,37)(16,38)(17,39)(22,29)(23,45)(24,46)(25,47)(27,43)(28,51)(32,54)(33,44)(35,55)(36,50)(40,58)(41,57)(42,59)(48,53)(49,62)(52,61)(56,63)(60,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,e^b=ehi,f^b=fh,d^c=dh,f^c=fi,e^d=ei,f^d=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, 53806);