#8758041

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

4626 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)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,19)(10,15,12,17)(13,21,22,18)(16,24,20,23)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(2,5)(6,7)(10,12)(13,22)(15,17)(18,21)(25,26,29,32)(27,33,35,30)(28,31,38,36)(34,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,24)(21,22)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,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, 8665237);