#8097717

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

4281 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004913);