#10451639

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

4337 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=gi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358835);