#7986219

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

4335 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=h,f^2=gi,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, 7893415);