#149229

order	512
nilpotency class	4
isoclinism	isoclinic class of #135851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11325
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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:

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

Polycyclic presentation:

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

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, 56425);