#8398674

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 8305870);