#149290

order	512
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11344
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

670 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56486);