#9009205

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4233 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22);(1,2,6,8)(3,5,7,4)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40);(1,3,6,7)(2,4,8,5)(11,16)(12,15)(13,21)(17,24)(26,30)(27,33)(28,35)(39,40);(1,4,6,5)(2,3,8,7)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,27,29,33)(26,31,30,37)(28,36,35,34)(32,40,38,39);(1,2)(3,4)(5,7)(6,8)(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,28,29,35)(26,32,30,38)(27,34,33,36)(31,39,37,40);(1,5,6,4)(2,7,8,3)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,e^2=gi,f^2=h,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, 8916401);