#10089205

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

4548 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)(11,20)(12,22)(13,19)(14,16)(15,17)(18,23)(21,24)(25,26)(27,37)(28,34)(29,30)(31,35)(32,36)(33,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,14)(10,16,17,20)(12,21,23,19)(13,18,24,22)(25,27,32,40)(26,33,36,37)(28,38,29,31)(30,39,34,35);(1,3)(2,5)(4,8)(6,7)(10,17)(12,23)(16,20)(19,21)(25,28,32,29)(26,34,36,30)(27,38,40,31)(33,35,37,39);(1,2)(3,5)(4,7)(6,8)(9,12)(10,18)(11,21)(13,20)(14,19)(15,23)(16,24)(17,22)(25,29,32,28)(26,34,36,30)(27,31,40,38)(33,35,37,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,24)(10,19,17,21)(11,22,14,18)(12,16,23,20)(25,30,32,34)(26,29,36,28)(27,39,40,35)(31,37,38,33);(1,2)(3,5)(4,7)(6,8)(9,14,15,11)(10,20,17,16)(12,19,23,21)(13,22,24,18)(25,31)(26,35)(27,28)(29,40)(30,33)(32,38)(34,37)(36,39)

Polycyclic presentation:

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

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, 9996401);