#9206466

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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)(18,22)(19,27)(20,30)(26,32)(35,38)(40,42)(43,45)(47,48);(2,7)(4,11)(8,9)(12,15)(17,18,23,22)(19,28,27,24)(20,25,30,31)(21,32,29,26)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,3,6,14)(2,8,7,9)(4,12,11,15)(5,13,10,16)(17,19,23,27)(18,24,22,28)(20,29,30,21)(25,32,31,26)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,9)(4,15)(5,13)(6,14)(7,8)(10,16)(11,12)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,4)(2,10)(3,12)(5,7)(6,11)(8,16)(9,13)(14,15)(17,21,23,29)(18,26,22,32)(19,20,27,30)(24,25,28,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,10)(2,11,7,4)(3,13,14,16)(8,15,9,12)(17,22,23,18)(19,28,27,24)(20,31,30,25)(21,32,29,26)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,e^2=i,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, 9113662);