#149384

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2092
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11377
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

427 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=ghi,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, 56580);