#486771

order	512
nilpotency class	4
isoclinism	isoclinic class of #486312
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	C₂²×C₈	C₂⁴
prime radical	#62384	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

677 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393967);