#488595

order	512
nilpotency class	4
isoclinism	isoclinic class of #488553
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
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 395791);