#10219809

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

4210 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)(11,20)(12,18)(13,23)(14,15)(16,19)(17,22)(21,24)(25,26,31,32)(27,28,29,33)(30,38,39,34)(35,40,37,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,21)(13,14)(15,18)(16,22)(19,20)(23,24)(26,32)(27,29)(34,38)(35,37);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,18,19,23)(11,21,22,14)(15,20,24,17)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,40,38,36);(2,6)(5,7)(9,13,16,12)(10,18,19,23)(11,21,22,14)(15,17,24,20)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(1,2)(3,5)(4,6)(7,8)(9,14,16,21)(10,15,19,24)(11,13,22,12)(17,18,20,23)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(1,3)(2,7)(4,8)(5,6)(9,15)(10,14)(11,18)(12,20)(13,17)(16,24)(19,21)(22,23)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=g,f^2=h,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, 10127005);