#8901315

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	#51104	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4350 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)(3,5)(4,8)(6,7)(25,26,30,31)(27,32,35,39)(28,29,38,36)(33,40,37,34);(2,7)(3,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26,30,31)(27,32,35,39)(28,36,38,29)(33,34,37,40);(1,3,6,8)(2,4,7,5)(10,14)(15,21)(16,22)(23,24)(26,31)(29,36)(32,39)(34,40);(1,4,6,5)(2,3,7,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,27,30,35)(26,32,31,39)(28,37,38,33)(29,34,36,40);(1,2)(3,4)(5,8)(6,7)(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,28)(26,29)(27,33)(30,38)(31,36)(32,40)(34,39)(35,37);(1,5,6,4)(2,8,7,3)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,29)(26,28)(27,34)(30,36)(31,38)(32,37)(33,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8808511);