#149144

order	512
nilpotency class	4
isoclinism	isoclinic class of #140094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1919
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

421 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)(3,16,21,45)(4,23,28,44)(5,13)(6,32)(7,34)(8,15)(9,18,38,53)(10,26,42,27)(11,30)(12,31)(14,33)(17,49)(19,46,20,39)(22,37,35,48)(24,57)(25,55,60,43)(29,41,61,56)(36,52)(40,50)(47,54)(51,63)(58,64)(59,62);(1,3)(2,9)(4,24,28,54)(5,19)(6,18)(7,35)(8,21)(10,40,42,62)(11,37)(12,16)(13,45)(14,46)(15,38)(17,27,52,26)(20,31)(22,33)(23,49,44,36)(25,51,60,58)(29,50,61,59)(30,53)(32,48)(34,39)(41,64,56,63)(43,57,55,47);(1,4,5,25,7,29,30,26,8,28,31,60,33,61,6,27)(2,10,11,41,14,43,13,23,15,42,32,56,34,55,12,44)(3,17,18,50,22,51,20,24,21,52,53,59,35,58,19,54)(9,36,16,47,39,63,48,62,38,49,45,57,46,64,37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,g^c=gi

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, 56340);