#8726312

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4253 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633508);