#489309

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61485	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(17,18,23,29)(19,30,33,42)(20,26,22,28)(21,24,38,41)(25,34,44,31)(27,35,40,36)(32,45,37,43)(39,47,46,48);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(18,24)(20,22)(21,36)(25,39)(26,27)(28,40)(29,41)(30,45)(31,34)(32,47)(35,38)(37,48)(42,43)(44,46);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,19,23,33)(18,25,29,44)(20,34,22,31)(21,37,38,32)(24,39,41,46)(26,30,28,42)(27,45,40,43)(35,47,36,48);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(17,20,23,22)(18,26,29,28)(19,31,33,34)(21,35,38,36)(24,40,41,27)(25,42,44,30)(32,47,37,48)(39,45,46,43)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396505);