#8445559

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

4440 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)(6,8)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(25,26,30,34)(27,29,33,31)(28,32,37,40)(35,36,39,38);(3,7)(6,8)(10,16)(11,12)(18,21)(19,20)(25,27,30,33)(26,29,34,31)(28,35,37,39)(32,36,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,27)(26,29)(28,35)(30,33)(31,34)(32,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,28,30,37)(26,32,34,40)(27,35,33,39)(29,36,31,38);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,29)(26,33)(27,34)(28,36)(30,31)(32,39)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8352755);