#148628

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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,g^2=i,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, 55824);