#147182

order	512
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10880
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 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)(3,16,20,45,19,44,52,33)(4,10,23,36,25,39,56,38)(5,13,7,11,28,41,29,42)(9,21,32,53,31,54,34,18)(17,46,49,61,48,43,63,64)(22,55,57,35,26,37,24,40)(30,58,60,51,59,50,62,47);(1,3,7,21,6,20,28,53,8,19,29,54,27,52,5,18)(2,9,13,33,14,34,42,44,15,31,41,45,12,32,11,16)(4,17,24,50,23,49,22,47,25,48,57,58,56,63,26,51)(10,30,37,61,38,62,35,46,39,59,55,64,36,60,40,43);(1,4,6,23,8,25,27,56)(2,10,12,36,15,39,14,38)(3,17,20,49,19,48,52,63)(5,26,7,24,28,22,29,57)(9,30,32,60,31,59,34,62)(11,40,41,55,42,35,13,37)(16,43,45,64,44,46,33,61)(18,51,21,50,53,47,54,58)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,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,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54378);