#10040791

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

4463 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947987);