#10173555

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

4484 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,18)(13,19)(14,24)(15,17)(20,21)(22,23)(26,32)(28,38)(33,35)(36,39);(3,8)(6,7)(9,11,15,14)(10,16,17,24)(12,20,19,22)(13,23,18,21)(25,26,31,32)(27,33,29,35)(28,34,38,30)(36,40,39,37);(1,3)(2,6)(4,7)(5,8)(10,17)(12,19)(16,24)(20,22)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,37,34,40);(3,8)(6,7)(9,12,15,19)(10,18,17,13)(11,20,14,22)(16,21,24,23)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,40)(35,37);(1,3)(2,7)(4,6)(5,8)(9,13,15,18)(10,19,17,12)(11,21,14,23)(16,20,24,22)(25,29,31,27)(26,35,32,33)(28,36,38,39)(30,37,34,40);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,16,17,24)(12,22,19,20)(13,23,18,21)(25,30,31,34)(26,28,32,38)(27,37,29,40)(33,36,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10080751);