#9057897

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

4529 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,22)(20,24)(21,23)(25,26,30,33)(27,28,31,34)(29,32,38,37)(35,36,39,40);(2,5)(6,7)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,14)(12,18)(13,21)(19,24)(25,26,30,33)(27,34,31,28)(29,37,38,32)(35,36,39,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,18)(10,16,14,11)(13,22,21,20)(17,19,23,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(9,13,15,21)(10,17,14,23)(11,19,16,24)(12,20,18,22)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(9,14,15,10)(11,18,16,12)(13,23,21,17)(19,22,24,20)(25,29,30,38)(26,32,33,37)(27,35,31,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965093);