#7655469

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

4308 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562665);