#8726651

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

4326 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)(3,7,8,5)(9,10,15,17)(11,19,13,16)(12,22,23,18)(14,20,24,21)(25,26,32,36)(27,31,29,34)(28,35,30,33)(37,40,38,39);(1,3,4,8)(2,5,6,7)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,18,24,22)(25,27)(26,31)(28,38)(29,32)(30,37)(33,40)(34,36)(35,39);(1,3)(2,5)(4,8)(6,7)(10,17)(12,23)(16,19)(20,21)(25,28,32,30)(26,33,36,35)(27,37,29,38)(31,39,34,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,19,24,16)(25,29,32,27)(26,34,36,31)(28,37,30,38)(33,39,35,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,19,17,16)(12,20,23,21)(14,18,24,22)(25,30,32,28)(26,35,36,33)(27,37,29,38)(31,39,34,40);(9,14,15,24)(10,20,17,21)(11,18,13,22)(12,16,23,19)(25,31,32,34)(26,29,36,27)(28,39,30,40)(33,38,35,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=gi,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, 8633847);