#8196946

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

4587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=ghi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104142);