#8993493

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

4524 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26)(27,29)(28,37)(30,38)(31,33)(32,36)(34,35)(39,40);(1,3,5,7)(2,6,4,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27,31,36)(26,32,33,29)(28,34,38,40)(30,35,37,39);(2,4)(3,7)(11,16)(12,15)(13,21)(17,24)(25,28,31,38)(26,30,33,37)(27,34,36,40)(29,39,32,35);(1,3,5,7)(2,6,4,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,28)(26,30)(27,34)(29,39)(31,38)(32,35)(33,37)(36,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,29,31,32)(26,27,33,36)(28,35,38,39)(30,40,37,34);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,30)(26,28)(27,35)(29,40)(31,37)(32,34)(33,38)(36,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8900689);