#148659

order	512
nilpotency class	4
isoclinism	isoclinic class of #13504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11200
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

631 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)(3,14,18,39)(4,20)(5,10)(6,28)(7,13)(8,16,34,19)(9,23)(11,26)(12,27)(15,46,21,56)(17,41,29,33)(22,42)(24,54)(25,55)(30,38)(31,40)(32,62,35,51)(36,58)(37,59)(43,60)(44,57,63,48)(45,50,61,52)(47,53,49,64);(1,3)(2,8)(4,21)(5,19)(6,29)(7,18)(9,35)(10,39)(11,41)(12,14)(13,34)(15,24)(16,27)(17,31)(20,53)(22,47)(23,48)(25,49)(26,50)(28,61)(30,51)(32,37)(33,43)(36,63)(38,44)(40,45)(42,46)(52,60)(54,64)(55,56)(57,59)(58,62);(1,4,7,24)(2,9)(3,15)(5,25,27,22)(6,30,31,58)(8,32,34,35)(10,38)(11,42)(12,36)(13,37)(14,44,39,63)(16,49)(17,51)(18,21)(19,47)(20,40)(23,26,59,60)(28,54)(29,62)(33,46,41,56)(43,55)(45,64,61,53)(48,50)(52,57)

Polycyclic presentation:

2^9|a^2=g,c^2=dh,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=gi,d^c=di,e^c=ehi,f^c=fi,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, 55855);