#146879

order	512
nilpotency class	4
isoclinism	isoclinic class of #128583
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10806
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

491 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)(3,15,19,34)(4,21)(5,11)(6,26)(7,13)(8,14)(9,17,33,20)(10,23)(12,28)(16,43,46,60)(18,48,50,32)(22,54)(24,55)(25,40)(27,35)(29,39)(30,38)(31,49,58,64)(36,57)(37,56)(41,62,59,45)(42,47,61,51)(44,63,52,53);(1,3,5,17,8,19,7,20)(2,9,13,34,14,33,11,15)(4,16,22,44,24,46,30,52)(6,18,25,47,29,50,28,51)(10,31,36,59,37,58,35,41)(12,32,39,61,40,48,26,42)(21,53,38,60,55,63,54,43)(23,45,27,49,56,62,57,64);(1,4,5,22,8,24,7,30)(2,10,11,35,14,37,13,36)(3,16,17,44,19,46,20,52)(6,27,25,56,29,57,28,23)(9,31,15,41,33,58,34,59)(12,38,26,21,40,54,39,55)(18,49,47,62,50,64,51,45)(32,60,42,53,48,43,61,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,f^b=fi,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54075);