#8700363

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

4440 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)(3,6)(4,7)(5,8)(10,12)(11,17)(16,20)(19,21)(25,26)(27,33)(28,39)(29,34)(30,38)(31,36)(32,37)(35,40);(1,3)(2,4)(5,7)(6,8)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(25,27,32,31)(26,33,37,36)(28,35,29,38)(30,39,40,34);(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,28,32,29)(26,34,37,39)(27,38,31,35)(30,33,40,36);(1,2,5,8)(3,6,7,4)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(25,29,32,28)(26,34,37,39)(27,38,31,35)(30,36,40,33);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,18,17,15)(13,16,22,20)(19,23,21,24)(25,30,32,40)(26,35,37,38)(27,39,31,34)(28,33,29,36);(1,4,5,6)(2,7,8,3)(9,13,14,22)(10,16,12,20)(11,19,17,21)(15,23,18,24)(25,31,32,27)(26,36,37,33)(28,35,29,38)(30,39,40,34)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8607559);