#8726618

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

4229 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)(3,7)(9,10,15,13)(11,16,19,12)(14,17,23,21)(18,22,24,20)(25,26,32,36)(27,31,29,34)(28,35,30,33)(37,40,38,39);(1,2,4,5)(3,6,7,8)(9,11,15,19)(10,16,13,12)(14,18,23,24)(17,22,21,20)(25,27)(26,31)(28,38)(29,32)(30,37)(33,40)(34,36)(35,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,16)(10,11,13,19)(14,20,23,22)(17,18,21,24)(25,28,32,30)(26,33,36,35)(27,37,29,38)(31,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,19)(14,22)(15,16)(17,24)(18,21)(20,23)(25,29,32,27)(26,34,36,31)(28,37,30,38)(33,39,35,40);(1,3,4,7)(2,6,5,8)(9,13,15,10)(11,12,19,16)(14,17,23,21)(18,22,24,20)(25,30,32,28)(26,35,36,33)(27,37,29,38)(31,39,34,40);(9,14,15,23)(10,17,13,21)(11,18,19,24)(12,20,16,22)(25,31,32,34)(26,29,36,27)(28,39,30,40)(33,38,35,37)

Polycyclic presentation:

2^9|a^2=gi,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, 8633814);