#8946565

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

4621 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=i,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, 8853761);