#8198193

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

4573 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:

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

Polycyclic presentation:

2^9|a^2=h,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,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8105389);