#8399479

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

4446 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,f^2=gi,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, 8306675);