#7656709

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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)(3,5,7,8)(9,10,16,15)(11,20,14,17)(12,22,21,18)(13,19,23,24)(27,33)(28,36)(30,34)(31,38);(1,2)(3,5)(4,6)(7,8)(9,11,16,14)(10,17,15,20)(12,13,21,23)(18,19,22,24)(25,26,29,32)(27,34,33,30)(28,31,36,38)(35,40,39,37);(9,12)(10,18)(11,13)(14,23)(15,22)(16,21)(17,19)(20,24)(25,27,29,33)(26,30,32,34)(28,35,36,39)(31,37,38,40);(1,3,4,7)(2,5,6,8)(9,13,16,23)(10,19,15,24)(11,21,14,12)(17,22,20,18)(25,28)(26,31)(27,35)(29,36)(30,37)(32,38)(33,39)(34,40);(9,14,16,11)(10,17,15,20)(12,23,21,13)(18,19,22,24);(9,15,16,10)(11,20,14,17)(12,22,21,18)(13,24,23,19)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=hi,f^2=hi,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, 7563905);