#8429537

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8336733);