#10218517

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

4333 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10125713);