#8969517

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

4361 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,e^2=gh,f^2=g,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, 8876713);