#7658088

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
prime radical	C₂²×#41	C₂³
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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565284);