#8759256

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

4668 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8666452);