#10201916

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

4406 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,e^2=gh,f^2=ghi,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, 10109112);