#147407

order	512
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×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	#10925
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

705 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,e^b=ei,f^b=fi,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, 54603);