#8933932

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

4420 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)(11,15)(12,13)(14,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,36)(28,34)(29,31)(30,37)(32,38)(33,39)(35,40);(3,7)(5,8)(9,11)(10,15)(12,23)(13,22)(14,20)(16,19)(17,24)(18,21)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,40)(37,39);(1,2)(3,5)(4,6)(7,8)(10,16)(11,20)(12,18)(22,24)(25,28,31,39)(26,33,29,34)(27,35,38,37)(30,32,40,36);(9,12)(10,17)(11,21)(13,16)(14,18)(15,22)(19,24)(20,23)(25,28)(26,34)(27,37)(29,33)(30,36)(31,39)(32,40)(35,38);(1,3,4,7)(2,5,6,8)(9,13,14,17)(10,18,16,12)(11,22,20,24)(15,21,19,23)(25,29,31,26)(27,36,38,32)(28,34,39,33)(30,37,40,35);(1,2)(3,5)(4,6)(7,8)(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,24)(18,23)(25,30,31,40)(26,35,29,37)(27,34,38,33)(28,36,39,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841128);