#8722290

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

4547 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,e^2=g,f^2=gi,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, 8629486);