#8463956

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

4491 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26,31,34)(27,30,33,28)(29,37,38,32)(35,36,40,39);(3,7)(6,8)(11,22)(12,23)(16,21)(17,24)(25,27,31,33)(26,28,34,30)(29,35,38,40)(32,36,37,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,20,21)(12,19,23,13)(14,17,18,24)(25,28,31,30)(26,27,34,33)(29,36,38,39)(32,35,37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,18)(20,24)(25,29)(26,32)(27,35)(28,36)(30,39)(31,38)(33,40)(34,37);(1,2)(3,6)(4,5)(7,8)(9,13,15,19)(10,18,20,14)(11,12,22,23)(16,24,21,17)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(9,14,15,18)(10,19,20,13)(11,17,22,24)(12,16,23,21)(25,30,31,28)(26,33,34,27)(29,36,38,39)(32,35,37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=i,f^2=gi,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, 8371152);