#7658876

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

4294 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)(9,10,16,15)(11,20,14,17)(12,18,21,23)(13,24,22,19)(26,28)(27,33)(32,36)(35,39);(1,3,5,8)(2,6,4,7)(9,11,16,14)(10,17,15,20)(12,22,21,13)(18,24,23,19)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,13)(14,22)(15,23)(16,21)(17,19)(20,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,2,5,4)(3,6,8,7)(9,13,16,22)(10,19,15,24)(11,21,14,12)(17,23,20,18)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,2,5,4)(3,6,8,7)(9,14,16,11)(10,17,15,20)(12,22,21,13)(18,19,23,24);(1,4,5,2)(3,7,8,6)(9,15,16,10)(11,20,14,17)(12,23,21,18)(13,24,22,19)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566072);