#10166319

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

4440 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)(4,6)(5,8)(9,10)(11,19)(12,22)(13,18)(14,16)(15,17)(20,24)(21,23)(25,26,31,30)(27,34,37,32)(28,33,29,38)(35,40,36,39);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,17,19)(12,20,18,23)(13,24,22,21)(25,27)(26,32)(28,35)(29,36)(30,34)(31,37)(33,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,17)(12,18)(16,19)(20,23)(25,28,31,29)(26,33,30,38)(27,35,37,36)(32,39,34,40);(1,3,4,8)(2,5,6,7)(9,12)(10,13)(11,20)(14,23)(15,18)(16,24)(17,22)(19,21)(25,29,31,28)(26,33,30,38)(27,36,37,35)(32,39,34,40);(9,13)(10,18)(11,21)(12,17)(14,24)(15,22)(16,20)(19,23)(25,30,31,26)(27,34,37,32)(28,38,29,33)(35,40,36,39);(9,14,15,11)(10,19,17,16)(12,23,18,20)(13,21,22,24)(25,27)(26,34)(28,35)(29,36)(30,32)(31,37)(33,40)(38,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,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, 10073515);