#10218556

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

4350 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10125752);