#8720385

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

4522 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)(3,4)(5,6)(7,8)(9,10)(11,18)(12,20)(13,16)(14,17)(15,21)(19,23)(22,24)(25,26)(27,32)(28,37)(29,34)(30,31)(33,38)(35,36)(39,40);(2,6)(3,7)(9,11,14,16)(10,13,17,18)(12,19,21,24)(15,23,20,22)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,38)(39,40);(1,3,5,7)(2,4,6,8)(12,21)(15,20)(19,24)(22,23)(26,31)(27,36)(33,37)(34,40);(1,4)(2,7)(3,6)(5,8)(9,12,14,21)(10,15,17,20)(11,19,16,24)(13,23,18,22)(25,27)(26,32)(28,34)(29,37)(30,36)(31,35)(33,39)(38,40);(9,10)(11,13)(12,15)(14,17)(16,18)(19,23)(20,21)(22,24)(25,28)(26,33)(27,34)(29,35)(30,38)(31,37)(32,39)(36,40);(9,13)(10,16)(11,17)(12,22)(14,18)(15,19)(20,24)(21,23)(25,29,30,39)(26,34,31,40)(27,33,36,37)(28,32,38,35)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627581);