#8186219

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

4552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8093415);