#10216869

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	#46703	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 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)(11,13)(12,16)(14,18)(15,22)(17,23)(19,24)(20,21)(25,26)(27,33)(28,29)(30,34)(31,40)(32,36)(35,37)(38,39);(1,3,6,5)(2,7,4,8)(9,11,14,17)(10,13,18,23)(12,20,22,19)(15,24,16,21)(25,27)(26,33)(28,35)(29,37)(30,40)(31,34)(32,38)(36,39);(9,12,14,22)(10,15,18,16)(11,19,17,20)(13,21,23,24)(25,28,32,30)(26,34,36,29)(27,35,38,40)(31,39,37,33);(2,4)(7,8)(9,12)(10,16)(11,20)(13,21)(14,22)(15,18)(17,19)(23,24)(25,29,32,34)(26,28,36,30)(27,37,38,31)(33,35,39,40);(1,4,6,2)(3,8,5,7)(9,11)(10,13)(12,19)(14,17)(15,21)(16,24)(18,23)(20,22)(25,30,32,28)(26,34,36,29)(27,35,38,40)(31,33,37,39);(1,5,6,3)(2,7,4,8)(9,13)(10,17)(11,18)(12,21)(14,23)(15,20)(16,19)(22,24)(25,31,32,37)(26,35,36,40)(27,34,38,29)(28,39,30,33)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=hi,e^2=ghi,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, 10124065);