#8942957

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

4496 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8850153);