#8636213

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4238 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8543409);