#9054386

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

4346 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)(3,4)(5,6)(7,8)(9,10,14,18)(11,16,15,12)(13,21,23,17)(19,20,24,22)(25,26)(27,29)(28,30)(31,33)(32,36)(34,38)(35,39)(37,40);(2,6)(3,7)(9,11)(10,12)(13,19)(14,15)(16,18)(17,20)(21,22)(23,24)(25,27,31,36)(26,32,33,29)(28,34,37,39)(30,35,40,38);(1,3,5,7)(2,4,6,8)(9,12,14,16)(10,15,18,11)(13,22,23,20)(17,19,21,24)(25,28,31,37)(26,30,33,40)(27,34,36,39)(29,38,32,35);(1,4)(2,7)(3,6)(5,8)(9,12)(10,11)(13,20)(14,16)(15,18)(17,19)(21,24)(22,23)(25,28)(26,30)(27,34)(29,38)(31,37)(32,35)(33,40)(36,39);(9,11)(10,16)(12,18)(13,19)(14,15)(17,22)(20,21)(23,24)(25,29,31,32)(26,27,33,36)(28,35,37,38)(30,39,40,34);(9,13,14,23)(10,17,18,21)(11,19,15,24)(12,20,16,22)(25,30)(26,28)(27,35)(29,39)(31,40)(32,34)(33,37)(36,38)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,e^2=ghi,f^2=h,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, 8961582);