#148812

order	512
nilpotency class	4
isoclinism	isoclinic class of #13469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2065
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11220
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

467 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fh,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, 56008);