#8874950

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

4458 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8782146);