#8932385

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

4713 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839581);