#489660

order	512
nilpotency class	4
isoclinism	isoclinic class of #489379
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	#33813	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

645 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)(19,29)(20,21)(22,39)(25,43)(26,27)(28,42)(30,34)(32,33)(35,36)(37,40)(41,46)(44,48)(45,47);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,18,23,28)(19,25,33,48)(20,24,36,42)(21,37,35,26)(22,40,39,27)(29,47,32,46)(30,45,34,41)(31,43,38,44);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,23,33)(18,25,28,48)(20,34,36,30)(21,38,35,31)(22,32,39,29)(24,41,42,45)(26,43,37,44)(27,47,40,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,35,23,36,39,21)(18,26,27,24,28,37,40,42)(19,30,32,38,33,34,29,31)(25,44,47,45,48,43,46,41)

Polycyclic presentation:

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

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, 396856);