#7981222

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

4499 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)(9,10,15,17)(11,19,13,16)(12,14,20,18)(21,22,23,24)(25,26,31,33)(27,36,39,32)(28,34,40,30)(29,37,38,35);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,24,18,22)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,36)(35,40);(1,3,5,4)(2,7,6,8)(10,17)(11,13)(12,20)(22,24)(26,33)(28,40)(32,36)(35,37);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,22,19,24)(25,28)(26,34)(27,37)(29,32)(30,33)(31,40)(35,39)(36,38);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,22,18,24)(25,29,31,38)(26,35,33,37)(27,34,39,30)(28,32,40,36);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,21,19,23)(25,30,31,34)(26,28,33,40)(27,38,39,29)(32,37,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=ghi,e^2=g,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, 7888418);