#9053884

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

4324 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)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(26,28)(27,34)(30,38)(37,39);(3,7)(5,8)(9,10,14,12)(11,15,18,17)(13,16,21,20)(19,22,24,23)(25,26,31,28)(27,29,34,32)(30,33,38,36)(35,37,40,39);(1,2)(3,5)(4,6)(7,8)(10,12)(11,18)(13,21)(22,23)(25,27,31,34)(26,32,28,29)(30,35,38,40)(33,39,36,37);(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(1,3)(2,5)(4,7)(6,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=g,e^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, 8961080);