#9206594

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	#58452	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=h,e^2=ghi,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, 9113790);