#8429701

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

4676 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8336897);