#8904491

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

4384 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,f^2=gh,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, 8811687);