#8689841

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

4567 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)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,7)(4,6)(5,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3)(2,5)(4,8)(6,7)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(2,6)(5,7)(9,12,15,10)(11,16,13,18)(14,17,22,20)(19,24,21,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3)(2,7)(4,8)(5,6)(9,13,15,11)(10,16,12,18)(14,19,22,21)(17,24,20,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=gh,f^2=g,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, 8597037);