#8943068

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4633 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)(11,19)(12,15)(13,17)(14,18)(16,22)(20,24)(21,23)(25,26,31,33)(27,37,40,32)(28,29,35,34)(30,38,39,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,31,40)(26,32,33,37)(28,38,35,36)(29,30,34,39);(1,2,4,6)(3,7,8,5)(13,21)(14,20)(17,23)(18,24)(26,33)(29,34)(30,39)(32,37);(3,8)(5,7)(9,12,15,10)(11,19,22,16)(13,17,21,23)(14,24,20,18)(25,28)(26,34)(27,38)(29,33)(30,37)(31,35)(32,39)(36,40);(1,3)(2,5)(4,8)(6,7)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,18,19,24)(25,29,31,34)(26,35,33,28)(27,39,40,30)(32,38,37,36);(1,2)(3,5)(4,6)(7,8)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,17,19,23)(25,30,31,39)(26,36,33,38)(27,34,40,29)(28,37,35,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8850264);