#148739

order	512
nilpotency class	4
isoclinism	isoclinic class of #136630
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11208
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

459 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)(3,14,18,34)(4,19)(5,10)(6,27)(7,12)(8,16,33,49)(9,22)(11,25)(13,26)(15,44,20,53)(17,32,28,38)(21,39)(23,57)(24,52)(29,41)(30,37)(31,60,62,50)(35,55)(36,56)(40,59)(42,54,64,46)(43,58,63,48)(45,51,47,61);(1,3,7,18)(2,8)(4,20)(5,16,26,49)(6,28,30,17)(9,31,36,62)(10,14)(11,38)(12,33)(13,34)(15,24)(19,51,52,61)(21,47)(22,46)(23,45)(25,58,59,48)(27,43)(29,50)(32,40)(35,42,41,64)(37,63)(39,53,57,44)(54,56)(55,60);(1,4,5,21,7,24,26,23)(2,9,10,35,12,36,13,41)(3,15,16,45,18,20,49,47)(6,29,25,22,30,55,59,56)(8,31,14,42,33,62,34,64)(11,39,37,52,40,57,27,19)(17,50,48,46,28,60,58,54)(32,44,43,51,38,53,63,61)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,e^c=eh,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, 55935);