#7974470

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

4559 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=h,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, 7881666);