#8393438

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

4515 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8300634);