#10166524

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

4466 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)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26)(27,35)(28,30)(29,32)(31,34)(33,40)(36,37)(38,39);(1,3,6,5)(2,4,7,8)(11,17)(14,21)(18,23)(22,24)(27,29)(32,35)(36,38)(37,39);(1,4)(2,5)(3,7)(6,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,37,40,39);(2,7)(4,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,39,35,37);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,29,31,27)(26,32,34,35)(28,36,33,38)(30,37,40,39);(1,5,6,3)(2,8,7,4)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,36)(35,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=i,f^2=g,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, 10073720);