#148747

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	#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)(4,18,21,52)(5,23)(6,28)(7,10)(9,20,34,55)(11,26)(12,30)(14,42)(15,45,24,57)(17,33,49,41)(19,53)(22,56,29,38)(25,35,60,39)(27,51)(31,43)(32,48,37,59)(36,46)(40,54)(44,62)(47,63)(50,64)(58,61);(1,3,6,17,7,16,30,49)(2,8,28,41,10,13,12,33)(4,19,22,50,21,14,29,44)(5,24,26,59,27,15,46,48)(9,31,39,63,34,40,35,61)(11,37,51,57,36,32,23,45)(18,42,56,62,52,53,38,64)(20,43,25,47,55,54,60,58);(1,4)(2,9,10,34)(3,14)(5,25)(6,29)(7,21)(8,31,13,40)(11,38,36,56)(12,39,28,35)(15,47)(16,19)(17,50)(18,51,52,23)(20,26)(22,30)(24,58)(27,60)(32,62,37,64)(33,63,41,61)(42,57,53,45)(43,48)(44,49)(46,55)(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,f^a=fi,h^a=hi,c^b=cf,e^b=ei,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, 55943);