#9206509

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	#52943	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=h,e^2=gi,f^2=gh,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, 9113705);