#8726529

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

4281 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=hi,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, 8633725);