#10451652

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

4276 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)(10,14)(11,17)(16,22)(19,20)(25,26,31,28)(27,34,33,30)(29,32,38,36)(35,40,39,37);(1,2,4,6)(3,5,8,7)(9,10,13,14)(11,15,17,18)(12,16,21,22)(19,23,20,24);(1,2)(3,5)(4,6)(7,8)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(3,8)(5,7)(9,11)(10,15)(12,20)(13,17)(14,18)(16,24)(19,21)(22,23)(25,28,31,26)(27,34,33,30)(29,32,38,36)(35,37,39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,23)(21,22)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,3)(2,5)(4,8)(6,7)(9,12,13,21)(10,16,14,22)(11,19,17,20)(15,23,18,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=hi,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, 10358848);