#8199504

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

4472 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)(11,16)(12,18)(13,20)(14,19)(15,17)(21,24)(22,23)(25,26,32,31)(27,35,30,33)(28,38,34,29)(36,37,39,40);(1,2)(3,7)(4,5)(6,8)(9,11)(10,16)(12,14)(13,17)(15,20)(18,19)(21,22)(23,24)(25,27,32,30)(26,33,31,35)(28,39,34,36)(29,40,38,37);(2,5)(6,7)(10,17)(11,20)(14,21)(18,23)(25,28)(26,29)(27,36)(30,39)(31,38)(32,34)(33,37)(35,40);(9,12)(10,18)(11,21)(13,19)(14,20)(15,22)(16,24)(17,23)(25,29,32,38)(26,34,31,28)(27,37,30,40)(33,39,35,36);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,19)(14,23)(15,16)(17,20)(18,21)(22,24)(25,30,32,27)(26,35,31,33)(28,39,34,36)(29,40,38,37);(1,2)(3,6)(4,5)(7,8)(9,14,15,21)(10,19,17,24)(11,12,20,22)(13,23,16,18)(25,31,32,26)(27,35,30,33)(28,38,34,29)(36,40,39,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=g,f^2=h,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, 8106700);