#8188170

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

4476 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)(11,19)(12,14)(13,16)(15,17)(18,23)(20,24)(21,22)(25,26)(27,28)(29,38)(30,33)(31,34)(32,35)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,17,19)(12,20,23,22)(14,21,18,24)(25,27)(26,29)(28,35)(30,36)(31,37)(32,38)(33,34)(39,40);(1,2,4,6)(3,5,8,7)(10,17)(14,18)(16,19)(21,24)(25,28,32,29)(26,27,35,38)(30,39,37,34)(31,33,36,40);(9,12)(10,18)(11,20)(13,22)(14,17)(15,23)(16,24)(19,21)(25,29,32,28)(26,27,35,38)(30,34,37,39)(31,33,36,40);(1,3)(2,5)(4,8)(6,7)(9,13,15,11)(10,19,17,16)(12,22,23,20)(14,24,18,21)(25,30,32,37)(26,33,35,40)(27,36,38,31)(28,39,29,34);(1,2)(3,5)(4,6)(7,8)(9,14,15,18)(10,12,17,23)(11,21,13,24)(16,20,19,22)(25,31,32,36)(26,34,35,39)(27,37,38,30)(28,40,29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095366);