#8900116

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

4477 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,e^2=gi,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, 8807312);