#8662896

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

4380 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570092);