#146613

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

952 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)(3,15,19,42,18,44,20,43)(4,21,23,39,24,10,28,35)(5,25,27,40,26,11,29,37)(9,30,33,17,32,48,38,49)(16,41,47,61,46,56,50,62)(22,34,53,59,52,36,54,60)(31,55,57,64,51,63,58,45);(1,3,7,20,8,18,6,19)(2,9,14,33,12,32,13,38)(4,16,23,47,24,46,28,50)(5,17,27,48,26,49,29,30)(10,31,35,58,21,51,39,57)(11,15,37,43,25,44,40,42)(22,45,54,63,52,64,53,55)(34,56,60,62,36,41,59,61);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,31)(11,36)(12,21)(13,39)(14,35)(15,41)(17,45)(18,46)(19,50)(20,47)(25,34)(26,52)(27,54)(29,53)(30,55)(32,51)(33,58)(37,59)(38,57)(40,60)(42,62)(43,61)(44,56)(48,63)(49,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,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, 53809);