#10504560

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

4634 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)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(26,30)(28,36)(31,33)(34,37);(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(26,30)(31,33)(32,38)(39,40);(1,2)(3,5)(4,6)(7,8)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(3,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27,29,35)(26,31,30,33)(28,37,36,34)(32,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,e^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, 10411756);