#8696023

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

4479 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)(3,7)(4,6)(5,8)(9,10,16,20)(11,21,23,17)(12,18,13,22)(14,15,19,24)(25,26,32,36)(27,37,39,33)(28,40,38,34)(29,31,30,35);(1,2,4,6)(3,5,8,7)(9,11,16,23)(10,17,20,21)(12,15,13,24)(14,22,19,18)(25,27)(26,33)(28,29)(30,38)(31,34)(32,39)(35,40)(36,37);(1,2)(3,5)(4,6)(7,8)(9,12,16,13)(10,18,20,22)(11,15,23,24)(14,21,19,17)(25,28,32,38)(26,34,36,40)(27,29,39,30)(31,37,35,33);(1,3,4,8)(2,5,6,7)(9,13,16,12)(10,18,20,22)(11,15,23,24)(14,17,19,21)(25,29,32,30)(26,35,36,31)(27,28,39,38)(33,40,37,34);(9,14)(10,15)(11,22)(12,21)(13,17)(16,19)(18,23)(20,24)(25,30,32,29)(26,31,36,35)(27,38,39,28)(33,34,37,40);(9,15)(10,19)(11,13)(12,23)(14,20)(16,24)(17,18)(21,22)(25,31)(26,30)(27,34)(28,37)(29,36)(32,35)(33,38)(39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=h,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, 8603219);