#149016

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	#11268
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

427 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)(3,15,19,40)(4,21)(5,12)(6,29)(7,32)(8,14)(9,17,35,26)(10,23)(11,27)(13,31)(16,45,49,33)(18,36,53,59)(20,34,52,44)(22,39)(24,41)(25,58)(28,42)(30,38)(37,57)(43,51,62,48)(46,55)(47,61,56,54)(50,60,64,63);(1,3,5,26,7,20,27,53,8,19,28,17,31,52,6,18)(2,9,29,15,13,36,42,44,14,35,11,40,32,59,12,34)(4,16,46,47,25,50,30,51,24,49,22,56,57,64,23,48)(10,33,37,62,39,60,41,61,38,45,58,43,55,63,21,54);(1,4,5,22,7,25,27,23,8,24,28,46,31,57,6,30)(2,10,11,37,13,39,12,41,14,38,29,58,32,55,42,21)(3,16,17,47,20,50,18,51,19,49,26,56,52,64,53,48)(9,33,15,43,36,60,44,54,35,45,40,62,59,63,34,61)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gi,e^c=ei,f^c=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, 56212);