#9027989

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

4261 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)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,26,31,36)(27,34,29,32)(28,38,37,33)(30,35,39,40);(1,2,5,4)(3,7,8,6)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27,31,29)(26,32,36,34)(28,39,37,30)(33,40,38,35);(1,3,5,8)(2,6,4,7)(9,10,14,18)(11,16,12,15)(13,21,22,17)(19,23,20,24)(25,28)(26,33)(27,30)(29,39)(31,37)(32,35)(34,40)(36,38);(3,8)(6,7)(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,29,31,27)(26,34,36,32)(28,39,37,30)(33,40,38,35);(1,3)(2,7)(4,6)(5,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,24)(20,23)(21,22)(25,30,31,39)(26,35,36,40)(27,37,29,28)(32,38,34,33);(1,4,5,2)(3,6,8,7)(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935185);