#8722181

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^2=g,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, 8629377);