#10166203

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

4440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=h,e^2=hi,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, 10073399);