#387442

order	512
nilpotency class	3
isoclinism	isoclinic class of #387441
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14473
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294638);