#8849992

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

4419 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8757188);