#10143361

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	#52005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,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, 10050557);