#8509778

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

4432 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416974);