#8917739

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

4333 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8824935);