#148745

order	512
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2101
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11208
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

463 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)(3,13,16,8)(4,18,21,51)(5,23)(6,27)(7,10)(9,20,34,54)(11,25)(12,29)(14,41)(15,44,46,56)(17,32,48,40)(19,52)(22,55,28,37)(24,35,58,38)(26,50)(30,42)(31,47,39,64)(33,53)(36,59)(43,61)(45,62)(49,63)(57,60);(1,3,6,17,7,16,29,48)(2,8,27,40,10,13,12,32)(4,19,22,49,21,14,28,43)(5,15,25,47,26,46,59,64)(9,33,38,60,34,30,35,62)(11,31,50,44,36,39,23,56)(18,41,55,61,51,52,37,63)(20,53,24,57,54,42,58,45);(1,4)(2,9,10,34)(3,14)(5,24)(6,28)(7,21)(8,30,13,33)(11,37,36,55)(12,38,27,35)(15,45)(16,19)(17,49)(18,50,51,23)(20,25)(22,29)(26,58)(31,61,39,63)(32,62,40,60)(41,56,52,44)(42,47)(43,48)(46,57)(53,64)(54,59)

Polycyclic presentation:

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

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, 55941);