#10143763

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=ghi,e^2=h,f^2=gh,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, 10050959);