#148275

order	512
nilpotency class	4
isoclinism	isoclinic class of #148273
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11087
commutator	C₂²×Q₈	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

774 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55471);