#8969526

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

4564 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,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, 8876722);