#8172638

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

4596 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8079834);