#7987959

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

4384 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7895155);