#8709060

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

4421 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)(11,19)(12,14)(13,16)(15,17)(18,23)(20,21)(22,24)(25,26)(27,31)(28,34)(29,33)(30,32)(35,38)(36,40)(37,39);(3,7)(6,8)(9,11,15,13)(10,16,17,19)(12,22,23,20)(14,21,18,24)(25,27,30,34)(26,31,32,28)(29,35,39,40)(33,38,37,36);(1,2)(3,6)(4,5)(7,8)(10,17)(12,23)(16,19)(20,22)(26,32)(27,34)(33,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,23)(10,18,17,14)(11,20,13,22)(16,21,19,24)(25,26)(27,28)(29,37)(30,32)(31,34)(33,39)(35,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,19,17,16)(12,22,23,20)(14,21,18,24)(25,28)(26,27)(29,38)(30,31)(32,34)(33,40)(35,37)(36,39);(9,14,15,18)(10,12,17,23)(11,21,13,24)(16,22,19,20)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,37)(34,40)

Polycyclic presentation:

2^9|b^2=gi,d^2=g,e^2=g,f^2=g,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, 8616256);