#9203773

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	#48083	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=i,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, 9110969);