#10143410

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

4300 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)(3,4)(5,7)(6,8)(9,10)(11,19)(12,22)(13,18)(14,16)(15,17)(20,24)(21,23)(25,26,31,29)(27,32,37,39)(28,34,38,33)(30,40,36,35);(1,3,6,5)(2,7,8,4)(9,11,15,14)(10,16,17,19)(12,20,18,23)(13,24,22,21)(25,27,31,37)(26,32,29,39)(28,36,38,30)(33,40,34,35);(10,17)(12,18)(16,19)(20,23)(25,28,31,38)(26,33,29,34)(27,30,37,36)(32,35,39,40);(1,4,6,7)(2,5,8,3)(9,12)(10,13)(11,20)(14,23)(15,18)(16,24)(17,22)(19,21)(25,28)(26,34)(27,36)(29,33)(30,37)(31,38)(32,35)(39,40);(1,5,6,3)(2,7,8,4)(9,13)(10,18)(11,21)(12,17)(14,24)(15,22)(16,20)(19,23)(25,29,31,26)(27,32,37,39)(28,34,38,33)(30,35,36,40);(1,2)(3,7)(4,5)(6,8)(9,14,15,11)(10,19,17,16)(12,23,18,20)(13,21,22,24)(25,30,31,36)(26,35,29,40)(27,28,37,38)(32,33,39,34)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=ghi,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, 10050606);