#8898279

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

4300 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)(11,18)(12,17)(13,19)(14,24)(15,16)(20,23)(21,22)(26,30)(31,36)(32,37)(38,40);(1,2)(3,5)(4,6)(7,8)(10,16)(13,24)(17,18)(20,22)(28,35)(32,37)(33,39)(38,40);(1,2,4,6)(3,5,7,8)(9,11,15,12)(10,17,16,18)(13,20,24,22)(14,23,19,21)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(3,7)(5,8)(9,12,15,11)(10,18,16,17)(13,20,24,22)(14,23,19,21)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,19)(11,20)(12,22)(14,16)(15,24)(17,21)(18,23)(25,28,29,35)(26,32,30,37)(27,33,34,39)(31,38,36,40);(1,2)(3,5)(4,6)(7,8)(9,14,15,19)(10,13,16,24)(11,21,12,23)(17,22,18,20)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40)

Polycyclic presentation:

2^9|c^2=gi,d^2=h,e^2=gh,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, 8805475);