#8621800

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	#46575	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4296 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8528996);