#8301429

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

4637 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)(12,14)(13,17)(15,19)(18,23)(20,21)(22,24)(25,26,32,31)(27,36,30,33)(28,35,29,34)(37,39,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(2,5)(6,8)(13,21)(17,20)(18,24)(22,23)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(3,7)(6,8)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,29,32,28)(26,35,31,34)(27,38,30,37)(33,40,36,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,30,32,27)(26,36,31,33)(28,38,29,37)(34,40,35,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,31,32,26)(27,36,30,33)(28,35,29,34)(37,40,38,39)

Polycyclic presentation:

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

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, 8208625);