#7901323

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

4472 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)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,24,20,22)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,24,21,22);(1,2,5,4)(3,6,8,7)(10,16)(11,12)(18,21)(19,20)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,22,21,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,24,17,22)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,4,5,2)(3,7,8,6)(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,20,21,19)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7808519);