#7932343

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

4295 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gh,f^2=hi,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, 7839539);