#8993343

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#50774	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8900539);