#10143750

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

4614 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,f^2=h,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, 10050946);