#10166060

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

4317 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)(11,21)(12,18)(13,14)(15,19)(16,20)(17,23)(22,24)(25,26,32,36)(27,31,29,34)(28,38,37,33)(30,35,39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,15)(13,22)(14,19)(16,23)(18,24)(20,21)(25,27,32,29)(26,31,36,34)(28,39,37,30)(33,35,38,40);(1,2,5,7)(3,6,8,4)(9,12,16,13)(10,18,20,14)(11,15,23,22)(17,24,21,19)(25,28,32,37)(26,33,36,38)(27,30,29,39)(31,40,34,35);(2,7)(4,6)(9,13,16,12)(10,18,20,14)(11,22,23,15)(17,24,21,19)(25,29,32,27)(26,34,36,31)(28,30,37,39)(33,40,38,35);(1,4)(2,3)(5,6)(7,8)(9,14,16,18)(10,13,20,12)(11,19,23,24)(15,17,22,21)(25,30)(26,35)(27,37)(28,29)(31,33)(32,39)(34,38)(36,40);(1,3)(2,6)(4,7)(5,8)(9,15,16,22)(10,19,20,24)(11,12,23,13)(14,21,18,17)(25,31,32,34)(26,27,36,29)(28,40,37,35)(30,38,39,33)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=gh,e^2=ghi,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, 10073256);