#8406583

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

4764 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313779);