#148806

order	512
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2085
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11220
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,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, 56002);