#148912

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2067
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11232
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 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)(3,17)(4,23,28,56)(5,29)(6,12)(7,15)(8,14)(9,16)(10,19)(11,26,42,59)(13,31)(18,35,52,63)(20,37)(21,44)(22,48)(24,57,53,39)(25,46,49,41)(27,60,34,40)(30,43)(32,45)(33,36)(38,54)(47,51,55,61)(50,62,58,64);(1,3,6,32,8,21,31,54,9,22,30,20,7,33,5,19)(2,10,29,36,15,37,43,48,16,38,13,44,14,45,12,17)(4,24,26,50,27,18,49,61,28,53,59,58,34,52,25,51)(11,39,56,47,41,35,60,62,42,57,23,55,46,63,40,64);(1,4,5,25,7,34,30,59,9,28,31,49,8,27,6,26)(2,11,12,40,14,46,13,23,16,42,43,60,15,41,29,56)(3,18,19,50,33,24,20,51,22,52,54,58,21,53,32,61)(10,35,17,47,45,39,44,64,38,63,48,55,37,57,36,62)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,f^c=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, 56108);