#8057915

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

4189 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

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, 7965111);