#7947535

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

4265 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,f^2=hi,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, 7854731);