#8720457

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	#46768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8627653);