#8689869

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

4609 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8597065);