#10216695

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	#59858	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4288 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)(12,15,21,24)(18,23,20,22)(25,26,32,36)(27,35,31,30)(28,37,39,33)(29,38,40,34);(1,3,6,5)(2,7,4,8)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,27,32,31)(26,30,36,35)(28,34,39,38)(29,37,40,33);(9,11,13,19)(10,14,16,17)(12,18,21,20)(15,22,24,23)(25,28,32,39)(26,33,36,37)(27,34,31,38)(29,35,40,30);(2,4)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24)(25,29,32,40)(26,34,36,38)(27,37,31,33)(28,30,39,35);(1,4,6,2)(3,8,5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,30)(26,27)(28,29)(31,36)(32,35)(33,34)(37,38)(39,40);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,31,32,27)(26,35,36,30)(28,38,39,34)(29,33,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=hi,e^2=ghi,f^2=gi,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, 10123891);