#9161731

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	#55697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4321 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9068927);