#8199209

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

4265 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:

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

Polycyclic presentation:

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

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, 8106405);