#7901359

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

4583 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)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26,31,30)(27,35,37,32)(28,38,39,33)(29,40,36,34);(1,3,5,8)(2,6,4,7)(25,27,31,37)(26,32,30,35)(28,36,39,29)(33,34,38,40);(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,28,31,39)(26,33,30,38)(27,29,37,36)(32,40,35,34);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28)(26,33)(27,36)(29,37)(30,38)(31,39)(32,34)(35,40);(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,29,31,36)(26,34,30,40)(27,28,37,39)(32,38,35,33);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,30,31,26)(27,35,37,32)(28,38,39,33)(29,34,36,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,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, 7808555);