#8689826

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

4525 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^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, 8597022);