#8395044

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

4559 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=gh,f^2=gh,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, 8302240);