#8726570

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gh,e^2=i,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, 8633766);