#8402596

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

4481 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309792);