#7912278

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

4432 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=hi,e^2=g,f^2=h,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, 7819474);