#147425

order	512
nilpotency class	4
isoclinism	isoclinic class of #13215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1961
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10927
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ch,e^b=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, 54621);