#8398615

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

4499 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:

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

Polycyclic presentation:

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

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, 8305811);