#8527469

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

4524 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)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26,32,31)(27,33,39,38)(28,37,29,34)(30,36,40,35);(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(17,21)(18,24)(25,27,32,39)(26,33,31,38)(28,36,29,35)(30,37,40,34);(1,2,4,6)(3,5,7,8)(10,15)(12,22)(16,19)(18,24)(25,28,32,29)(26,34,31,37)(27,35,39,36)(30,33,40,38);(3,7)(5,8)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,23)(22,24)(25,29,32,28)(26,34,31,37)(27,36,39,35)(30,33,40,38);(1,2)(3,5)(4,6)(7,8)(9,12,14,22)(10,17,15,21)(11,18,20,24)(13,19,23,16)(25,30,32,40)(26,35,31,36)(27,37,39,34)(28,38,29,33);(1,3)(2,5)(4,7)(6,8)(9,13)(10,18)(11,21)(12,19)(14,23)(15,24)(16,22)(17,20)(25,31,32,26)(27,38,39,33)(28,37,29,34)(30,36,40,35)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8434665);