#10143315

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

4299 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)(3,7)(4,5)(6,8)(9,10,15,17)(11,16,21,23)(12,18,20,14)(13,19,22,24)(25,26)(27,34)(28,33)(29,30)(31,35)(32,39)(36,38)(37,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,22)(13,20)(14,19)(15,21)(17,23)(18,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,39)(33,40)(34,35);(1,3)(2,6)(4,7)(5,8)(10,17)(11,21)(12,20)(19,24)(25,28,31,29)(26,33,35,30)(27,36,39,37)(32,40,34,38);(1,3,5,8)(2,6,4,7)(9,12,15,20)(10,18,17,14)(11,22,21,13)(16,24,23,19)(25,29,31,28)(26,33,35,30)(27,37,39,36)(32,40,34,38);(1,3)(2,7)(4,6)(5,8)(9,13,15,22)(10,19,17,24)(11,12,21,20)(14,16,18,23)(25,30,31,33)(26,29,35,28)(27,38,39,40)(32,37,34,36);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,20,17,12)(11,19,21,24)(13,23,22,16)(25,27)(26,34)(28,36)(29,37)(30,40)(31,39)(32,35)(33,38)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=gi,e^2=gh,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, 10050511);