#8429442

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8336638);