#8614168

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8521364);