#149116

order	512
nilpotency class	4
isoclinism	isoclinic class of #140091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1918
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,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, 56312);