#147197

order	512
nilpotency class	4
isoclinism	isoclinic class of #13291
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1993
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10881
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

489 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)(3,16,20,48,19,47,30,37)(4,11,24,41,27,43,26,60)(5,14,7,31,29,45,28,12)(10,21,36,18,35,54,38,53)(17,49,22,56,51,46,57,64)(23,44,32,61,59,42,25,40)(34,52,63,58,62,55,39,50);(1,3,7,21,6,30,29,53,9,19,28,54,8,20,5,18)(2,10,14,37,33,36,12,16,15,35,45,48,13,38,31,47)(4,22,25,58,24,17,23,52,27,57,32,50,26,51,59,55)(11,39,42,46,60,34,61,64,43,63,44,49,41,62,40,56);(1,4,6,24,9,27,8,26)(2,11,13,41,15,43,33,60)(3,17,20,22,19,51,30,57)(5,23,7,32,29,59,28,25)(10,34,36,63,35,62,38,39)(12,40,14,44,31,61,45,42)(16,46,48,64,47,49,37,56)(18,50,54,52,53,58,21,55)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,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, 54393);