#7973832

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

4485 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)(11,16)(12,21)(13,14)(15,17)(18,24)(19,23)(20,22)(25,26,30,34)(27,35,36,31)(28,32,39,38)(29,33,40,37);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,21)(13,19,24,22)(14,23,18,20)(25,27,30,36)(26,31,34,35)(28,37,39,33)(29,32,40,38);(1,2,5,4)(3,6,8,7)(10,17)(11,12)(13,24)(20,23)(27,36)(28,39)(31,35)(32,38);(9,12,15,11)(10,16,17,21)(13,22,24,19)(14,23,18,20)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(9,13,15,24)(10,18,17,14)(11,19,12,22)(16,20,21,23)(25,28)(26,32)(27,37)(29,35)(30,39)(31,40)(33,36)(34,38);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,13,17,24)(11,20,12,23)(16,22,21,19)(25,29,30,40)(26,33,34,37)(27,38,36,32)(28,31,39,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=h,e^2=h,f^2=gh,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, 7881028);