#9206486

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	#48561	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,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, 9113682);