#10480772

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

4419 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)(12,20)(15,21)(18,24)(22,23)(26,31)(27,32)(29,37)(36,40);(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40);(3,7)(5,8)(9,12,13,20)(10,15,17,21)(11,18,19,24)(14,22,16,23)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,29,30,37)(26,33,31,38)(27,35,32,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10387968);