#148627

order	512
nilpotency class	4
isoclinism	isoclinic class of #135291
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1825
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11199
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

623 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)(3,14,18,37)(4,20,24,54)(5,10)(6,28)(7,12)(8,16,34,19)(9,23,36,57)(11,26)(13,27)(15,45)(17,39,29,33)(21,55)(22,53,25,40)(30,42,58,35)(31,38)(32,61)(41,59)(43,56)(44,48,60,50)(46,52)(47,63)(49,64)(51,62);(1,3)(2,8)(4,21)(5,19)(6,29)(7,18)(9,32)(10,37)(11,39)(12,34)(13,14)(15,24)(16,27)(17,31)(20,52)(22,46)(23,47)(25,51)(26,48)(28,60)(30,49)(33,41)(35,63)(36,64)(38,44)(40,55)(42,43)(45,53)(50,59)(54,62)(56,57)(58,61);(1,4,5,22,7,24,27,25)(2,9,10,35,12,36,13,42)(3,15,16,46,18,21,19,51)(6,30,26,57,31,58,59,23)(8,32,14,43,34,64,37,63)(11,40,38,20,41,53,28,54)(17,49,48,47,29,61,50,56)(33,45,44,52,39,55,60,62)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=gi,e^c=ehi,f^c=fi,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, 55823);