#7645814

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

4366 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)(10,14)(12,20)(15,17)(18,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,36)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(1,4,5,3)(2,8,6,7)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,38)(26,32,33,37)(27,35,31,39)(28,36,34,40)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553010);