#9206591

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3872 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)(3,9,14,8)(4,10,11,5)(12,13,15,16)(17,18,23,21)(19,26,20,24)(22,25,30,29)(27,32,28,31)(34,38)(35,43)(36,46)(42,48);(2,7)(4,11)(8,9)(12,15)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,34,39,38)(35,44,43,40)(36,41,46,47)(37,48,45,42);(1,3,6,14)(2,8,7,9)(4,12,11,15)(5,13,10,16)(17,18,23,21)(19,24,20,26)(22,25,30,29)(27,31,28,32)(33,35,39,43)(34,40,38,44)(36,45,46,37)(41,48,47,42);(1,3)(2,9)(4,15)(5,13)(6,14)(7,8)(10,16)(11,12)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,4)(2,10)(3,12)(5,7)(6,11)(8,16)(9,13)(14,15)(17,21,23,18)(19,26,20,24)(22,25,30,29)(27,31,28,32)(33,37,39,45)(34,42,38,48)(35,36,43,46)(40,41,44,47);(1,5,6,10)(2,11,7,4)(3,13,14,16)(8,15,9,12)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,34)(35,44,43,40)(36,47,46,41)(37,48,45,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,e^2=ghi,f^2=h,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, 9113787);