#8891770

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

4430 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)(11,18)(12,17)(13,22)(14,15)(16,21)(19,24)(20,23)(25,26,30,31)(27,32,35,39)(28,29,38,36)(33,40,37,34);(1,2)(3,6)(4,5)(7,8)(10,15)(12,22)(16,18)(19,23)(25,26,30,31)(27,32,35,39)(28,36,38,29)(33,34,37,40);(2,5)(6,8)(10,15)(13,17)(16,18)(20,24)(26,31)(29,36)(32,39)(34,40);(3,7)(6,8)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,27,30,35)(26,32,31,39)(28,37,38,33)(29,34,36,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,28)(26,29)(27,33)(30,38)(31,36)(32,40)(34,39)(35,37);(1,2)(3,6)(4,5)(7,8)(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,29)(26,28)(27,34)(30,36)(31,38)(32,37)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8798966);