#9206560

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3887 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)(3,8)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18,22,23)(19,25,30,24)(20,26,27,21)(28,29,31,32)(34,38)(35,43)(36,46)(42,48);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(18,23)(20,27)(24,25)(28,31)(33,34,39,38)(35,44,43,40)(36,41,46,47)(37,48,45,42);(2,4)(3,9)(10,12)(11,16)(17,19,22,30)(18,24,23,25)(20,28,27,31)(21,29,26,32)(33,35,39,43)(34,40,38,44)(36,45,46,37)(41,48,47,42);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,19)(18,25)(20,31)(21,29)(22,30)(23,24)(26,32)(27,28)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,20)(18,26)(19,28)(21,23)(22,27)(24,32)(25,29)(30,31)(33,37,39,45)(34,42,38,48)(35,36,43,46)(40,41,44,47);(1,6,7,15)(2,10,4,12)(3,11,9,16)(5,13,8,14)(17,21,22,26)(18,27,23,20)(19,29,30,32)(24,31,25,28)(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=i,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, 9113756);