#149126

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2067
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
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

409 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)(4,24,29,45)(5,13)(6,11)(7,34)(8,15)(9,19)(10,27,42,26)(12,33)(14,17)(18,51,53,56)(20,39)(21,38)(22,37)(23,49)(25,40,58,64)(28,57,60,43)(30,61,50,41)(31,47)(32,46)(35,44)(36,59,63,54)(48,52,62,55);(1,3,17,22,8,23,7,35)(2,9,14,47,15,39,34,38)(4,25,30,55,29,58,50,52)(5,31,33,20,32,21,6,19)(10,40,57,48,42,64,43,62)(11,44,13,16,12,37,46,49)(18,26,59,28,53,27,54,60)(24,56,61,36,45,51,41,63);(1,4,5,26,17,50,33,60,8,29,32,27,7,30,6,28)(2,10,11,24,34,57,46,61,15,42,12,45,14,43,13,41)(3,18,19,52,35,59,21,25,23,53,20,55,22,54,31,58)(9,36,16,48,47,56,37,40,39,63,49,62,38,51,44,64)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,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, 56322);