#7974482

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

4580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=g,e^2=gh,f^2=g,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, 7881678);