#488712

order	512
nilpotency class	4
isoclinism	isoclinic class of #488707
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 ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

749 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)(19,31)(20,34)(21,37)(22,29)(23,28)(24,30)(25,32)(26,36)(27,40)(33,45)(35,44)(38,47)(39,42)(41,46)(43,48);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,24,20)(18,25,30,44)(21,38,41,48)(22,33,23,36)(26,42,45,27)(28,43,29,47)(31,46,34,37)(32,39,35,40);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20,24,19)(18,26,30,45)(21,35,41,32)(22,33,23,36)(25,42,44,27)(28,31,29,34)(37,43,46,47)(38,40,48,39);(2,8)(6,13)(7,14)(12,15)(17,21,22,39,24,41,23,40)(18,27,28,46,30,42,29,37)(19,32,36,48,20,35,33,38)(25,31,47,45,44,34,43,26)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395908);