#149223

order	512
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2085
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:

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

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,e^b=ei,g^b=gi,f^c=fi,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, 56419);