#10219717

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

4355 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)(11,16)(12,19)(13,21)(14,15)(17,23)(18,20)(22,24)(25,26,32,36)(27,37,40,33)(28,34,29,39)(30,35,38,31);(1,2)(3,7)(4,6)(5,8)(9,11,14,19)(10,12,15,16)(13,18,23,22)(17,20,21,24)(25,27)(26,33)(28,38)(29,30)(31,39)(32,40)(34,35)(36,37);(1,3)(2,5)(4,8)(6,7)(10,15)(12,16)(17,21)(20,24)(25,28,32,29)(26,34,36,39)(27,38,40,30)(31,33,35,37);(2,6)(5,7)(9,12)(10,11)(13,20)(14,16)(15,19)(17,18)(21,22)(23,24)(25,29,32,28)(26,34,36,39)(27,38,40,30)(31,37,35,33);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,15)(13,22)(14,19)(17,20)(18,23)(21,24)(25,30,32,38)(26,35,36,31)(27,29,40,28)(33,34,37,39);(1,3)(2,7)(4,8)(5,6)(9,13,14,23)(10,17,15,21)(11,18,19,22)(12,20,16,24)(25,31,32,35)(26,30,36,38)(27,39,40,34)(28,33,29,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^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, 10126913);