#8391335

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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)(11,19)(12,20)(13,18)(14,16)(15,21)(17,23)(22,24)(25,26,31,33)(27,34,28,30)(29,37,39,32)(35,38,36,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,21)(10,15,16,19)(12,18,23,22)(13,20,24,17)(25,26,31,33)(27,34,28,30)(29,37,39,32)(35,38,36,40);(2,5)(6,7)(10,16)(13,24)(15,19)(17,20)(25,27,31,28)(26,30,33,34)(29,35,39,36)(32,38,37,40);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,18)(13,15)(14,23)(16,20)(19,24)(21,22)(25,28,31,27)(26,30,33,34)(29,35,39,36)(32,40,37,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,24)(18,23)(25,29,31,39)(26,32,33,37)(27,35,28,36)(30,38,34,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,16,22)(11,20,21,17)(12,15,23,19)(25,30)(26,27)(28,33)(29,38)(31,34)(32,35)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=i,e^2=gh,f^2=g,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, 8298531);