#8599479

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

4473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8506675);