#9160803

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=hi,e^2=h,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^d=egi,f^e=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, 9067999);