#8704203

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

4461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8611399);