#8617788

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

4438 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)(9,10,15,19)(11,16,13,17)(12,14,18,24)(20,21,22,23)(25,26)(27,31)(28,38)(29,30)(32,33)(34,39)(35,36)(37,40);(2,5)(6,7)(9,11)(10,16)(12,20)(13,15)(14,21)(17,19)(18,22)(23,24)(25,27,32,37)(26,31,33,40)(28,35,29,34)(30,39,38,36);(1,2)(3,6)(4,5)(7,8)(11,13)(16,17)(20,22)(21,23)(25,28,32,29)(26,30,33,38)(27,34,37,35)(31,36,40,39);(1,3,4,8)(2,6,5,7)(9,12)(10,14)(11,20)(13,22)(15,18)(16,21)(17,23)(19,24)(25,29,32,28)(26,30,33,38)(27,35,37,34)(31,36,40,39);(9,13,15,11)(10,17,19,16)(12,22,18,20)(14,23,24,21)(25,30)(26,28)(27,36)(29,33)(31,34)(32,38)(35,40)(37,39);(9,14)(10,18)(11,21)(12,19)(13,23)(15,24)(16,22)(17,20)(25,31,32,40)(26,27,33,37)(28,36,29,39)(30,34,38,35)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8524984);