#147185

order	512
nilpotency class	4
isoclinism	isoclinic class of #13077
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:

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

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=gh,h^a=hi,d^b=di,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, 54381);