#8197044

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

4544 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:

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

Polycyclic presentation:

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

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, 8104240);