#9189555

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	#46544	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9096751);