#147005

order	512
nilpotency class	4
isoclinism	isoclinic class of #13215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×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	#10843
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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)(3,16,20,47,19,46,53,34)(4,10,24,38,26,41,59,40)(5,13,7,11,28,42,30,43)(9,21,33,54,32,55,35,18)(17,48,22,57,51,45,58,64)(23,44,29,37,49,39,25,60)(31,52,63,50,62,56,36,61);(1,3,7,21,6,20,28,54,8,19,30,55,27,53,5,18)(2,9,13,34,14,35,43,46,15,32,42,47,12,33,11,16)(4,22,25,50,24,51,23,56,26,58,29,61,59,17,49,52)(10,36,39,45,40,62,37,57,41,63,44,48,38,31,60,64);(1,4)(2,10)(3,17,19,51)(5,23)(6,24)(7,29)(8,26)(9,31,32,62)(11,37)(12,38)(13,44)(14,40)(15,41)(16,45,46,48)(18,50,54,61)(20,22,53,58)(21,56,55,52)(25,30)(27,59)(28,49)(33,63,35,36)(34,57,47,64)(39,42)(43,60)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,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, 54201);