#8531677

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

4576 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8438873);