#8132464

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

4288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8039660);