#9027223

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

4453 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)(3,8)(4,7)(5,6)(9,10,14,17)(11,15,19,13)(12,16,22,24)(18,23,20,21)(25,26)(27,29)(28,37)(30,38)(31,33)(32,36)(34,35)(39,40);(3,4)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,22)(21,24)(25,27,31,36)(26,32,33,29)(28,34,38,40)(30,35,37,39);(1,3)(2,7)(4,6)(5,8)(9,11,14,19)(10,15,17,13)(12,20,22,18)(16,21,24,23)(25,28,31,38)(26,30,33,37)(27,34,36,40)(29,39,32,35);(3,4)(7,8)(9,11)(10,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24)(25,28)(26,30)(27,34)(29,39)(31,38)(32,35)(33,37)(36,40);(1,4,6,3)(2,7,5,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,29,31,32)(26,27,33,36)(28,35,38,39)(30,40,37,34);(1,5,6,2)(3,7,4,8)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,24,20)(25,30)(26,28)(27,35)(29,40)(31,37)(32,34)(33,38)(36,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,e^2=h,f^2=i,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, 8934419);