#8135661

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

4578 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8042857);