#10143584

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
prime radical	#50656	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4328 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)(9,10)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26,32,36)(27,33,38,40)(28,34,29,31)(30,35,37,39);(3,4)(7,8)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(25,27,32,38)(26,33,36,40)(28,39,29,35)(30,31,37,34);(1,3,6,4)(2,7,5,8)(10,17)(11,20)(12,22)(18,24)(25,28,32,29)(26,34,36,31)(27,35,38,39)(30,33,37,40);(3,4)(7,8)(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,29,32,28)(26,31,36,34)(27,35,38,39)(30,33,37,40);(1,4,6,3)(2,7,5,8)(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,30)(26,35)(27,31)(28,33)(29,40)(32,37)(34,38)(36,39);(1,5,6,2)(3,8,4,7)(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,31)(26,28)(27,37)(29,36)(30,38)(32,34)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=hi,e^2=gh,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, 10050780);