#8726342

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=g,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, 8633538);