#8172745

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

4301 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,e^2=g,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, 8079941);