#8935416

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

4466 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842612);