#8186221

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
prime radical	#38950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 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)(25,26,31,34)(27,33,29,32)(28,37,39,30)(35,36,38,40);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,24)(21,23)(25,27,31,29)(26,32,34,33)(28,35,39,38)(30,36,37,40);(2,5)(6,7)(9,11,15,12)(10,16,18,14)(13,19,23,20)(17,24,21,22)(25,28,31,39)(26,30,34,37)(27,35,29,38)(32,36,33,40);(1,3,4,8)(2,6,5,7)(9,12,15,11)(10,16,18,14)(13,20,23,19)(17,24,21,22)(25,27,31,29)(26,32,34,33)(28,35,39,38)(30,36,37,40);(9,13,15,23)(10,17,18,21)(11,19,12,20)(14,22,16,24)(25,29,31,27)(26,33,34,32)(28,38,39,35)(30,40,37,36);(1,2)(3,6)(4,5)(7,8)(9,14,15,16)(10,11,18,12)(13,22,23,24)(17,19,21,20)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^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, 8093417);