#147411

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

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

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=ghi,h^a=hi,d^b=di,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, 54607);