#8607449

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	#50522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8514645);