#10143575

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

4280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=ghi,e^2=h,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, 10050771);