#489669

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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	#33737	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

644 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(18,24)(19,28)(20,31)(21,22)(25,39)(26,43)(27,38)(29,45)(30,35)(32,36)(33,34)(37,40)(41,48)(42,46);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,18)(19,25)(20,32)(21,26)(22,36)(23,27)(24,31)(28,40)(29,46)(30,42)(33,38)(34,43)(35,41)(37,44)(39,45)(47,48);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,23,30)(18,25,27,42)(20,29,34,45)(21,35,22,28)(24,37,38,48)(26,41,36,40)(31,44,33,47)(32,46,43,39);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,21,33,23,34,22,31)(18,24,36,43,27,38,26,32)(19,29,35,47,30,45,28,44)(25,37,40,39,42,48,41,46)

Polycyclic presentation:

2^9|c^2=hi,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396865);