#244560

order	512
nilpotency class	3
isoclinism	isoclinic class of #244547
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1017 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fh,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151756);